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Theorem sticksstones2 41270
Description: The range function on strictly monotone functions with finite domain and codomain is an injective mapping onto 𝐾-elemental sets. (Contributed by metakunt, 27-Sep-2024.)
Hypotheses
Ref Expression
sticksstones2.1 (πœ‘ β†’ 𝑁 ∈ β„•0)
sticksstones2.2 (πœ‘ β†’ 𝐾 ∈ β„•0)
sticksstones2.3 𝐡 = {π‘Ž ∈ 𝒫 (1...𝑁) ∣ (β™―β€˜π‘Ž) = 𝐾}
sticksstones2.4 𝐴 = {𝑓 ∣ (𝑓:(1...𝐾)⟢(1...𝑁) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘“β€˜π‘₯) < (π‘“β€˜π‘¦)))}
sticksstones2.5 𝐹 = (𝑧 ∈ 𝐴 ↦ ran 𝑧)
Assertion
Ref Expression
sticksstones2 (πœ‘ β†’ 𝐹:𝐴–1-1→𝐡)
Distinct variable groups:   𝐴,π‘Ž,𝑧   𝐴,𝑓,𝑧   𝑧,𝐡   𝐾,π‘Ž,π‘₯,𝑦   𝑓,𝐾,π‘₯,𝑦   𝑁,π‘Ž   𝑓,𝑁   πœ‘,π‘Ž,𝑧   πœ‘,𝑓   π‘₯,𝑧,𝑦
Allowed substitution hints:   πœ‘(π‘₯,𝑦)   𝐴(π‘₯,𝑦)   𝐡(π‘₯,𝑦,𝑓,π‘Ž)   𝐹(π‘₯,𝑦,𝑧,𝑓,π‘Ž)   𝐾(𝑧)   𝑁(π‘₯,𝑦,𝑧)

Proof of Theorem sticksstones2
Dummy variables 𝑏 𝑖 𝑗 π‘Ÿ 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveqeq2 6900 . . . . . 6 (π‘Ž = ran 𝑧 β†’ ((β™―β€˜π‘Ž) = 𝐾 ↔ (β™―β€˜ran 𝑧) = 𝐾))
2 fzfid 13943 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ 𝐴) β†’ (1...𝑁) ∈ Fin)
3 eleq1w 2815 . . . . . . . . . . . . 13 (𝑓 = 𝑧 β†’ (𝑓 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
4 feq1 6698 . . . . . . . . . . . . . 14 (𝑓 = 𝑧 β†’ (𝑓:(1...𝐾)⟢(1...𝑁) ↔ 𝑧:(1...𝐾)⟢(1...𝑁)))
5 fveq1 6890 . . . . . . . . . . . . . . . . . 18 (𝑓 = 𝑧 β†’ (π‘“β€˜π‘₯) = (π‘§β€˜π‘₯))
6 fveq1 6890 . . . . . . . . . . . . . . . . . 18 (𝑓 = 𝑧 β†’ (π‘“β€˜π‘¦) = (π‘§β€˜π‘¦))
75, 6breq12d 5161 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝑧 β†’ ((π‘“β€˜π‘₯) < (π‘“β€˜π‘¦) ↔ (π‘§β€˜π‘₯) < (π‘§β€˜π‘¦)))
87imbi2d 340 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑧 β†’ ((π‘₯ < 𝑦 β†’ (π‘“β€˜π‘₯) < (π‘“β€˜π‘¦)) ↔ (π‘₯ < 𝑦 β†’ (π‘§β€˜π‘₯) < (π‘§β€˜π‘¦))))
98ralbidv 3176 . . . . . . . . . . . . . . 15 (𝑓 = 𝑧 β†’ (βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘“β€˜π‘₯) < (π‘“β€˜π‘¦)) ↔ βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘§β€˜π‘₯) < (π‘§β€˜π‘¦))))
109ralbidv 3176 . . . . . . . . . . . . . 14 (𝑓 = 𝑧 β†’ (βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘“β€˜π‘₯) < (π‘“β€˜π‘¦)) ↔ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘§β€˜π‘₯) < (π‘§β€˜π‘¦))))
114, 10anbi12d 630 . . . . . . . . . . . . 13 (𝑓 = 𝑧 β†’ ((𝑓:(1...𝐾)⟢(1...𝑁) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘“β€˜π‘₯) < (π‘“β€˜π‘¦))) ↔ (𝑧:(1...𝐾)⟢(1...𝑁) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘§β€˜π‘₯) < (π‘§β€˜π‘¦)))))
123, 11bibi12d 345 . . . . . . . . . . . 12 (𝑓 = 𝑧 β†’ ((𝑓 ∈ 𝐴 ↔ (𝑓:(1...𝐾)⟢(1...𝑁) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘“β€˜π‘₯) < (π‘“β€˜π‘¦)))) ↔ (𝑧 ∈ 𝐴 ↔ (𝑧:(1...𝐾)⟢(1...𝑁) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘§β€˜π‘₯) < (π‘§β€˜π‘¦))))))
13 sticksstones2.4 . . . . . . . . . . . . . 14 𝐴 = {𝑓 ∣ (𝑓:(1...𝐾)⟢(1...𝑁) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘“β€˜π‘₯) < (π‘“β€˜π‘¦)))}
14 eqabb 2872 . . . . . . . . . . . . . 14 (𝐴 = {𝑓 ∣ (𝑓:(1...𝐾)⟢(1...𝑁) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘“β€˜π‘₯) < (π‘“β€˜π‘¦)))} ↔ βˆ€π‘“(𝑓 ∈ 𝐴 ↔ (𝑓:(1...𝐾)⟢(1...𝑁) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘“β€˜π‘₯) < (π‘“β€˜π‘¦)))))
1513, 14mpbi 229 . . . . . . . . . . . . 13 βˆ€π‘“(𝑓 ∈ 𝐴 ↔ (𝑓:(1...𝐾)⟢(1...𝑁) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘“β€˜π‘₯) < (π‘“β€˜π‘¦))))
1615spi 2176 . . . . . . . . . . . 12 (𝑓 ∈ 𝐴 ↔ (𝑓:(1...𝐾)⟢(1...𝑁) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘“β€˜π‘₯) < (π‘“β€˜π‘¦))))
1712, 16chvarvv 2001 . . . . . . . . . . 11 (𝑧 ∈ 𝐴 ↔ (𝑧:(1...𝐾)⟢(1...𝑁) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘§β€˜π‘₯) < (π‘§β€˜π‘¦))))
1817biimpi 215 . . . . . . . . . 10 (𝑧 ∈ 𝐴 β†’ (𝑧:(1...𝐾)⟢(1...𝑁) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘§β€˜π‘₯) < (π‘§β€˜π‘¦))))
1918adantl 481 . . . . . . . . 9 ((πœ‘ ∧ 𝑧 ∈ 𝐴) β†’ (𝑧:(1...𝐾)⟢(1...𝑁) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘§β€˜π‘₯) < (π‘§β€˜π‘¦))))
2019simpld 494 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ 𝐴) β†’ 𝑧:(1...𝐾)⟢(1...𝑁))
2120frnd 6725 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ 𝐴) β†’ ran 𝑧 βŠ† (1...𝑁))
222, 21sselpwd 5326 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ 𝐴) β†’ ran 𝑧 ∈ 𝒫 (1...𝑁))
2320ffnd 6718 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑧 ∈ 𝐴) β†’ 𝑧 Fn (1...𝐾))
24 hashfn 14340 . . . . . . . . . . 11 (𝑧 Fn (1...𝐾) β†’ (β™―β€˜π‘§) = (β™―β€˜(1...𝐾)))
2523, 24syl 17 . . . . . . . . . 10 ((πœ‘ ∧ 𝑧 ∈ 𝐴) β†’ (β™―β€˜π‘§) = (β™―β€˜(1...𝐾)))
26 sticksstones2.2 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐾 ∈ β„•0)
2726adantr 480 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑧 ∈ 𝐴) β†’ 𝐾 ∈ β„•0)
28 hashfz1 14311 . . . . . . . . . . 11 (𝐾 ∈ β„•0 β†’ (β™―β€˜(1...𝐾)) = 𝐾)
2927, 28syl 17 . . . . . . . . . 10 ((πœ‘ ∧ 𝑧 ∈ 𝐴) β†’ (β™―β€˜(1...𝐾)) = 𝐾)
3025, 29eqtrd 2771 . . . . . . . . 9 ((πœ‘ ∧ 𝑧 ∈ 𝐴) β†’ (β™―β€˜π‘§) = 𝐾)
3130eqcomd 2737 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ 𝐴) β†’ 𝐾 = (β™―β€˜π‘§))
32 fzfid 13943 . . . . . . . . 9 ((πœ‘ ∧ 𝑧 ∈ 𝐴) β†’ (1...𝐾) ∈ Fin)
33 elfznn 13535 . . . . . . . . . . . . . . . . . . . 20 (π‘Ž ∈ (1...𝐾) β†’ π‘Ž ∈ β„•)
34333ad2ant3 1134 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) β†’ π‘Ž ∈ β„•)
3534nnred 12232 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) β†’ π‘Ž ∈ ℝ)
3635adantr 480 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) β†’ π‘Ž ∈ ℝ)
37 elfznn 13535 . . . . . . . . . . . . . . . . . . 19 (𝑏 ∈ (1...𝐾) β†’ 𝑏 ∈ β„•)
3837nnred 12232 . . . . . . . . . . . . . . . . . 18 (𝑏 ∈ (1...𝐾) β†’ 𝑏 ∈ ℝ)
3938adantl 481 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) β†’ 𝑏 ∈ ℝ)
40 lttri2 11301 . . . . . . . . . . . . . . . . 17 ((π‘Ž ∈ ℝ ∧ 𝑏 ∈ ℝ) β†’ (π‘Ž β‰  𝑏 ↔ (π‘Ž < 𝑏 ∨ 𝑏 < π‘Ž)))
4136, 39, 40syl2anc 583 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) β†’ (π‘Ž β‰  𝑏 ↔ (π‘Ž < 𝑏 ∨ 𝑏 < π‘Ž)))
42203adant3 1131 . . . . . . . . . . . . . . . . . . . . . . . 24 ((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) β†’ 𝑧:(1...𝐾)⟢(1...𝑁))
43 simp3 1137 . . . . . . . . . . . . . . . . . . . . . . . 24 ((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) β†’ π‘Ž ∈ (1...𝐾))
4442, 43ffvelcdmd 7087 . . . . . . . . . . . . . . . . . . . . . . 23 ((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) β†’ (π‘§β€˜π‘Ž) ∈ (1...𝑁))
4544adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) β†’ (π‘§β€˜π‘Ž) ∈ (1...𝑁))
4645adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ π‘Ž < 𝑏) β†’ (π‘§β€˜π‘Ž) ∈ (1...𝑁))
47 elfznn 13535 . . . . . . . . . . . . . . . . . . . . 21 ((π‘§β€˜π‘Ž) ∈ (1...𝑁) β†’ (π‘§β€˜π‘Ž) ∈ β„•)
4846, 47syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ π‘Ž < 𝑏) β†’ (π‘§β€˜π‘Ž) ∈ β„•)
4948nnred 12232 . . . . . . . . . . . . . . . . . . 19 ((((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ π‘Ž < 𝑏) β†’ (π‘§β€˜π‘Ž) ∈ ℝ)
5019simprd 495 . . . . . . . . . . . . . . . . . . . . . . 23 ((πœ‘ ∧ 𝑧 ∈ 𝐴) β†’ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘§β€˜π‘₯) < (π‘§β€˜π‘¦)))
51503adant3 1131 . . . . . . . . . . . . . . . . . . . . . 22 ((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) β†’ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘§β€˜π‘₯) < (π‘§β€˜π‘¦)))
5251adantr 480 . . . . . . . . . . . . . . . . . . . . 21 (((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) β†’ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘§β€˜π‘₯) < (π‘§β€˜π‘¦)))
5343adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) β†’ π‘Ž ∈ (1...𝐾))
54 simpr 484 . . . . . . . . . . . . . . . . . . . . . 22 (((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) β†’ 𝑏 ∈ (1...𝐾))
55 breq1 5151 . . . . . . . . . . . . . . . . . . . . . . . 24 (π‘₯ = π‘Ž β†’ (π‘₯ < 𝑦 ↔ π‘Ž < 𝑦))
56 fveq2 6891 . . . . . . . . . . . . . . . . . . . . . . . . 25 (π‘₯ = π‘Ž β†’ (π‘§β€˜π‘₯) = (π‘§β€˜π‘Ž))
5756breq1d 5158 . . . . . . . . . . . . . . . . . . . . . . . 24 (π‘₯ = π‘Ž β†’ ((π‘§β€˜π‘₯) < (π‘§β€˜π‘¦) ↔ (π‘§β€˜π‘Ž) < (π‘§β€˜π‘¦)))
5855, 57imbi12d 344 . . . . . . . . . . . . . . . . . . . . . . 23 (π‘₯ = π‘Ž β†’ ((π‘₯ < 𝑦 β†’ (π‘§β€˜π‘₯) < (π‘§β€˜π‘¦)) ↔ (π‘Ž < 𝑦 β†’ (π‘§β€˜π‘Ž) < (π‘§β€˜π‘¦))))
59 breq2 5152 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑏 β†’ (π‘Ž < 𝑦 ↔ π‘Ž < 𝑏))
60 fveq2 6891 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = 𝑏 β†’ (π‘§β€˜π‘¦) = (π‘§β€˜π‘))
6160breq2d 5160 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑏 β†’ ((π‘§β€˜π‘Ž) < (π‘§β€˜π‘¦) ↔ (π‘§β€˜π‘Ž) < (π‘§β€˜π‘)))
6259, 61imbi12d 344 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = 𝑏 β†’ ((π‘Ž < 𝑦 β†’ (π‘§β€˜π‘Ž) < (π‘§β€˜π‘¦)) ↔ (π‘Ž < 𝑏 β†’ (π‘§β€˜π‘Ž) < (π‘§β€˜π‘))))
6358, 62rspc2v 3622 . . . . . . . . . . . . . . . . . . . . . 22 ((π‘Ž ∈ (1...𝐾) ∧ 𝑏 ∈ (1...𝐾)) β†’ (βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘§β€˜π‘₯) < (π‘§β€˜π‘¦)) β†’ (π‘Ž < 𝑏 β†’ (π‘§β€˜π‘Ž) < (π‘§β€˜π‘))))
6453, 54, 63syl2anc 583 . . . . . . . . . . . . . . . . . . . . 21 (((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) β†’ (βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘§β€˜π‘₯) < (π‘§β€˜π‘¦)) β†’ (π‘Ž < 𝑏 β†’ (π‘§β€˜π‘Ž) < (π‘§β€˜π‘))))
6552, 64mpd 15 . . . . . . . . . . . . . . . . . . . 20 (((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) β†’ (π‘Ž < 𝑏 β†’ (π‘§β€˜π‘Ž) < (π‘§β€˜π‘)))
6665imp 406 . . . . . . . . . . . . . . . . . . 19 ((((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ π‘Ž < 𝑏) β†’ (π‘§β€˜π‘Ž) < (π‘§β€˜π‘))
6749, 66ltned 11355 . . . . . . . . . . . . . . . . . 18 ((((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ π‘Ž < 𝑏) β†’ (π‘§β€˜π‘Ž) β‰  (π‘§β€˜π‘))
6842ffvelcdmda 7086 . . . . . . . . . . . . . . . . . . . . . . 23 (((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) β†’ (π‘§β€˜π‘) ∈ (1...𝑁))
69 elfznn 13535 . . . . . . . . . . . . . . . . . . . . . . 23 ((π‘§β€˜π‘) ∈ (1...𝑁) β†’ (π‘§β€˜π‘) ∈ β„•)
7068, 69syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) β†’ (π‘§β€˜π‘) ∈ β„•)
7170nnred 12232 . . . . . . . . . . . . . . . . . . . . 21 (((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) β†’ (π‘§β€˜π‘) ∈ ℝ)
7271adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑏 < π‘Ž) β†’ (π‘§β€˜π‘) ∈ ℝ)
73 breq1 5151 . . . . . . . . . . . . . . . . . . . . . . . . 25 (π‘₯ = 𝑏 β†’ (π‘₯ < 𝑦 ↔ 𝑏 < 𝑦))
74 fveq2 6891 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (π‘₯ = 𝑏 β†’ (π‘§β€˜π‘₯) = (π‘§β€˜π‘))
7574breq1d 5158 . . . . . . . . . . . . . . . . . . . . . . . . 25 (π‘₯ = 𝑏 β†’ ((π‘§β€˜π‘₯) < (π‘§β€˜π‘¦) ↔ (π‘§β€˜π‘) < (π‘§β€˜π‘¦)))
7673, 75imbi12d 344 . . . . . . . . . . . . . . . . . . . . . . . 24 (π‘₯ = 𝑏 β†’ ((π‘₯ < 𝑦 β†’ (π‘§β€˜π‘₯) < (π‘§β€˜π‘¦)) ↔ (𝑏 < 𝑦 β†’ (π‘§β€˜π‘) < (π‘§β€˜π‘¦))))
77 breq2 5152 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = π‘Ž β†’ (𝑏 < 𝑦 ↔ 𝑏 < π‘Ž))
78 fveq2 6891 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = π‘Ž β†’ (π‘§β€˜π‘¦) = (π‘§β€˜π‘Ž))
7978breq2d 5160 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = π‘Ž β†’ ((π‘§β€˜π‘) < (π‘§β€˜π‘¦) ↔ (π‘§β€˜π‘) < (π‘§β€˜π‘Ž)))
8077, 79imbi12d 344 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = π‘Ž β†’ ((𝑏 < 𝑦 β†’ (π‘§β€˜π‘) < (π‘§β€˜π‘¦)) ↔ (𝑏 < π‘Ž β†’ (π‘§β€˜π‘) < (π‘§β€˜π‘Ž))))
8176, 80rspc2v 3622 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑏 ∈ (1...𝐾) ∧ π‘Ž ∈ (1...𝐾)) β†’ (βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘§β€˜π‘₯) < (π‘§β€˜π‘¦)) β†’ (𝑏 < π‘Ž β†’ (π‘§β€˜π‘) < (π‘§β€˜π‘Ž))))
8254, 53, 81syl2anc 583 . . . . . . . . . . . . . . . . . . . . . 22 (((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) β†’ (βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘§β€˜π‘₯) < (π‘§β€˜π‘¦)) β†’ (𝑏 < π‘Ž β†’ (π‘§β€˜π‘) < (π‘§β€˜π‘Ž))))
8352, 82mpd 15 . . . . . . . . . . . . . . . . . . . . 21 (((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) β†’ (𝑏 < π‘Ž β†’ (π‘§β€˜π‘) < (π‘§β€˜π‘Ž)))
8483imp 406 . . . . . . . . . . . . . . . . . . . 20 ((((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑏 < π‘Ž) β†’ (π‘§β€˜π‘) < (π‘§β€˜π‘Ž))
8572, 84ltned 11355 . . . . . . . . . . . . . . . . . . 19 ((((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑏 < π‘Ž) β†’ (π‘§β€˜π‘) β‰  (π‘§β€˜π‘Ž))
8685necomd 2995 . . . . . . . . . . . . . . . . . 18 ((((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ 𝑏 < π‘Ž) β†’ (π‘§β€˜π‘Ž) β‰  (π‘§β€˜π‘))
8767, 86jaodan 955 . . . . . . . . . . . . . . . . 17 ((((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) ∧ (π‘Ž < 𝑏 ∨ 𝑏 < π‘Ž)) β†’ (π‘§β€˜π‘Ž) β‰  (π‘§β€˜π‘))
8887ex 412 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) β†’ ((π‘Ž < 𝑏 ∨ 𝑏 < π‘Ž) β†’ (π‘§β€˜π‘Ž) β‰  (π‘§β€˜π‘)))
8941, 88sylbid 239 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) β†’ (π‘Ž β‰  𝑏 β†’ (π‘§β€˜π‘Ž) β‰  (π‘§β€˜π‘)))
9089necon4d 2963 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) ∧ 𝑏 ∈ (1...𝐾)) β†’ ((π‘§β€˜π‘Ž) = (π‘§β€˜π‘) β†’ π‘Ž = 𝑏))
9190ralrimiva 3145 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑧 ∈ 𝐴 ∧ π‘Ž ∈ (1...𝐾)) β†’ βˆ€π‘ ∈ (1...𝐾)((π‘§β€˜π‘Ž) = (π‘§β€˜π‘) β†’ π‘Ž = 𝑏))
92913expa 1117 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑧 ∈ 𝐴) ∧ π‘Ž ∈ (1...𝐾)) β†’ βˆ€π‘ ∈ (1...𝐾)((π‘§β€˜π‘Ž) = (π‘§β€˜π‘) β†’ π‘Ž = 𝑏))
9392ralrimiva 3145 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑧 ∈ 𝐴) β†’ βˆ€π‘Ž ∈ (1...𝐾)βˆ€π‘ ∈ (1...𝐾)((π‘§β€˜π‘Ž) = (π‘§β€˜π‘) β†’ π‘Ž = 𝑏))
9420, 93jca 511 . . . . . . . . . 10 ((πœ‘ ∧ 𝑧 ∈ 𝐴) β†’ (𝑧:(1...𝐾)⟢(1...𝑁) ∧ βˆ€π‘Ž ∈ (1...𝐾)βˆ€π‘ ∈ (1...𝐾)((π‘§β€˜π‘Ž) = (π‘§β€˜π‘) β†’ π‘Ž = 𝑏)))
95 dff13 7257 . . . . . . . . . 10 (𝑧:(1...𝐾)–1-1β†’(1...𝑁) ↔ (𝑧:(1...𝐾)⟢(1...𝑁) ∧ βˆ€π‘Ž ∈ (1...𝐾)βˆ€π‘ ∈ (1...𝐾)((π‘§β€˜π‘Ž) = (π‘§β€˜π‘) β†’ π‘Ž = 𝑏)))
9694, 95sylibr 233 . . . . . . . . 9 ((πœ‘ ∧ 𝑧 ∈ 𝐴) β†’ 𝑧:(1...𝐾)–1-1β†’(1...𝑁))
97 hashf1rn 14317 . . . . . . . . 9 (((1...𝐾) ∈ Fin ∧ 𝑧:(1...𝐾)–1-1β†’(1...𝑁)) β†’ (β™―β€˜π‘§) = (β™―β€˜ran 𝑧))
9832, 96, 97syl2anc 583 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ 𝐴) β†’ (β™―β€˜π‘§) = (β™―β€˜ran 𝑧))
9931, 98eqtrd 2771 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ 𝐴) β†’ 𝐾 = (β™―β€˜ran 𝑧))
10099eqcomd 2737 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ 𝐴) β†’ (β™―β€˜ran 𝑧) = 𝐾)
1011, 22, 100elrabd 3685 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ 𝐴) β†’ ran 𝑧 ∈ {π‘Ž ∈ 𝒫 (1...𝑁) ∣ (β™―β€˜π‘Ž) = 𝐾})
102 sticksstones2.3 . . . . . . 7 𝐡 = {π‘Ž ∈ 𝒫 (1...𝑁) ∣ (β™―β€˜π‘Ž) = 𝐾}
103102eleq2i 2824 . . . . . 6 (ran 𝑧 ∈ 𝐡 ↔ ran 𝑧 ∈ {π‘Ž ∈ 𝒫 (1...𝑁) ∣ (β™―β€˜π‘Ž) = 𝐾})
104103a1i 11 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ 𝐴) β†’ (ran 𝑧 ∈ 𝐡 ↔ ran 𝑧 ∈ {π‘Ž ∈ 𝒫 (1...𝑁) ∣ (β™―β€˜π‘Ž) = 𝐾}))
105101, 104mpbird 257 . . . 4 ((πœ‘ ∧ 𝑧 ∈ 𝐴) β†’ ran 𝑧 ∈ 𝐡)
106 sticksstones2.5 . . . 4 𝐹 = (𝑧 ∈ 𝐴 ↦ ran 𝑧)
107105, 106fmptd 7115 . . 3 (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
108 sticksstones2.1 . . . . . . . . . . . 12 (πœ‘ β†’ 𝑁 ∈ β„•0)
1091083ad2ant1 1132 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) β†’ 𝑁 ∈ β„•0)
110109adantr 480 . . . . . . . . . 10 (((πœ‘ ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 β‰  𝑗) β†’ 𝑁 ∈ β„•0)
111263ad2ant1 1132 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) β†’ 𝐾 ∈ β„•0)
112111adantr 480 . . . . . . . . . 10 (((πœ‘ ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 β‰  𝑗) β†’ 𝐾 ∈ β„•0)
113 simpl2 1191 . . . . . . . . . 10 (((πœ‘ ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 β‰  𝑗) β†’ 𝑖 ∈ 𝐴)
114 simpl3 1192 . . . . . . . . . 10 (((πœ‘ ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 β‰  𝑗) β†’ 𝑗 ∈ 𝐴)
115 simpr 484 . . . . . . . . . 10 (((πœ‘ ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 β‰  𝑗) β†’ 𝑖 β‰  𝑗)
116 fveq2 6891 . . . . . . . . . . . . 13 (π‘Ÿ = 𝑠 β†’ (π‘–β€˜π‘Ÿ) = (π‘–β€˜π‘ ))
117 fveq2 6891 . . . . . . . . . . . . 13 (π‘Ÿ = 𝑠 β†’ (π‘—β€˜π‘Ÿ) = (π‘—β€˜π‘ ))
118116, 117neeq12d 3001 . . . . . . . . . . . 12 (π‘Ÿ = 𝑠 β†’ ((π‘–β€˜π‘Ÿ) β‰  (π‘—β€˜π‘Ÿ) ↔ (π‘–β€˜π‘ ) β‰  (π‘—β€˜π‘ )))
119118cbvrabv 3441 . . . . . . . . . . 11 {π‘Ÿ ∈ (1...𝐾) ∣ (π‘–β€˜π‘Ÿ) β‰  (π‘—β€˜π‘Ÿ)} = {𝑠 ∈ (1...𝐾) ∣ (π‘–β€˜π‘ ) β‰  (π‘—β€˜π‘ )}
120119infeq1i 9477 . . . . . . . . . 10 inf({π‘Ÿ ∈ (1...𝐾) ∣ (π‘–β€˜π‘Ÿ) β‰  (π‘—β€˜π‘Ÿ)}, ℝ, < ) = inf({𝑠 ∈ (1...𝐾) ∣ (π‘–β€˜π‘ ) β‰  (π‘—β€˜π‘ )}, ℝ, < )
121110, 112, 13, 113, 114, 115, 120sticksstones1 41269 . . . . . . . . 9 (((πœ‘ ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 β‰  𝑗) β†’ ran 𝑖 β‰  ran 𝑗)
122106a1i 11 . . . . . . . . . 10 (((πœ‘ ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 β‰  𝑗) β†’ 𝐹 = (𝑧 ∈ 𝐴 ↦ ran 𝑧))
123 simpr 484 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 β‰  𝑗) ∧ 𝑧 = 𝑖) β†’ 𝑧 = 𝑖)
124123rneqd 5937 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 β‰  𝑗) ∧ 𝑧 = 𝑖) β†’ ran 𝑧 = ran 𝑖)
125 fzfid 13943 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 β‰  𝑗) β†’ (1...𝑁) ∈ Fin)
126 eleq1w 2815 . . . . . . . . . . . . . . . . . . 19 (𝑓 = 𝑖 β†’ (𝑓 ∈ 𝐴 ↔ 𝑖 ∈ 𝐴))
127 feq1 6698 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = 𝑖 β†’ (𝑓:(1...𝐾)⟢(1...𝑁) ↔ 𝑖:(1...𝐾)⟢(1...𝑁)))
128 fveq1 6890 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓 = 𝑖 β†’ (π‘“β€˜π‘₯) = (π‘–β€˜π‘₯))
129 fveq1 6890 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓 = 𝑖 β†’ (π‘“β€˜π‘¦) = (π‘–β€˜π‘¦))
130128, 129breq12d 5161 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 = 𝑖 β†’ ((π‘“β€˜π‘₯) < (π‘“β€˜π‘¦) ↔ (π‘–β€˜π‘₯) < (π‘–β€˜π‘¦)))
131130imbi2d 340 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = 𝑖 β†’ ((π‘₯ < 𝑦 β†’ (π‘“β€˜π‘₯) < (π‘“β€˜π‘¦)) ↔ (π‘₯ < 𝑦 β†’ (π‘–β€˜π‘₯) < (π‘–β€˜π‘¦))))
1321312ralbidv 3217 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = 𝑖 β†’ (βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘“β€˜π‘₯) < (π‘“β€˜π‘¦)) ↔ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘–β€˜π‘₯) < (π‘–β€˜π‘¦))))
133127, 132anbi12d 630 . . . . . . . . . . . . . . . . . . 19 (𝑓 = 𝑖 β†’ ((𝑓:(1...𝐾)⟢(1...𝑁) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘“β€˜π‘₯) < (π‘“β€˜π‘¦))) ↔ (𝑖:(1...𝐾)⟢(1...𝑁) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘–β€˜π‘₯) < (π‘–β€˜π‘¦)))))
134126, 133bibi12d 345 . . . . . . . . . . . . . . . . . 18 (𝑓 = 𝑖 β†’ ((𝑓 ∈ 𝐴 ↔ (𝑓:(1...𝐾)⟢(1...𝑁) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘“β€˜π‘₯) < (π‘“β€˜π‘¦)))) ↔ (𝑖 ∈ 𝐴 ↔ (𝑖:(1...𝐾)⟢(1...𝑁) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘–β€˜π‘₯) < (π‘–β€˜π‘¦))))))
135134, 16chvarvv 2001 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ 𝐴 ↔ (𝑖:(1...𝐾)⟢(1...𝑁) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘–β€˜π‘₯) < (π‘–β€˜π‘¦))))
136135biimpi 215 . . . . . . . . . . . . . . . 16 (𝑖 ∈ 𝐴 β†’ (𝑖:(1...𝐾)⟢(1...𝑁) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘–β€˜π‘₯) < (π‘–β€˜π‘¦))))
137136adantl 481 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝑖 ∈ 𝐴) β†’ (𝑖:(1...𝐾)⟢(1...𝑁) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘–β€˜π‘₯) < (π‘–β€˜π‘¦))))
138137simpld 494 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑖 ∈ 𝐴) β†’ 𝑖:(1...𝐾)⟢(1...𝑁))
1391383adant3 1131 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) β†’ 𝑖:(1...𝐾)⟢(1...𝑁))
140139adantr 480 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 β‰  𝑗) β†’ 𝑖:(1...𝐾)⟢(1...𝑁))
141140frnd 6725 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 β‰  𝑗) β†’ ran 𝑖 βŠ† (1...𝑁))
142125, 141sselpwd 5326 . . . . . . . . . 10 (((πœ‘ ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 β‰  𝑗) β†’ ran 𝑖 ∈ 𝒫 (1...𝑁))
143122, 124, 113, 142fvmptd 7005 . . . . . . . . 9 (((πœ‘ ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 β‰  𝑗) β†’ (πΉβ€˜π‘–) = ran 𝑖)
144 simpr 484 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 β‰  𝑗) ∧ 𝑧 = 𝑗) β†’ 𝑧 = 𝑗)
145144rneqd 5937 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 β‰  𝑗) ∧ 𝑧 = 𝑗) β†’ ran 𝑧 = ran 𝑗)
146 fzfid 13943 . . . . . . . . . . . . 13 (πœ‘ β†’ (1...𝑁) ∈ Fin)
1471463ad2ant1 1132 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) β†’ (1...𝑁) ∈ Fin)
148 eleq1w 2815 . . . . . . . . . . . . . . . . . . 19 (𝑓 = 𝑗 β†’ (𝑓 ∈ 𝐴 ↔ 𝑗 ∈ 𝐴))
149 feq1 6698 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = 𝑗 β†’ (𝑓:(1...𝐾)⟢(1...𝑁) ↔ 𝑗:(1...𝐾)⟢(1...𝑁)))
150 fveq1 6890 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓 = 𝑗 β†’ (π‘“β€˜π‘₯) = (π‘—β€˜π‘₯))
151 fveq1 6890 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓 = 𝑗 β†’ (π‘“β€˜π‘¦) = (π‘—β€˜π‘¦))
152150, 151breq12d 5161 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 = 𝑗 β†’ ((π‘“β€˜π‘₯) < (π‘“β€˜π‘¦) ↔ (π‘—β€˜π‘₯) < (π‘—β€˜π‘¦)))
153152imbi2d 340 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = 𝑗 β†’ ((π‘₯ < 𝑦 β†’ (π‘“β€˜π‘₯) < (π‘“β€˜π‘¦)) ↔ (π‘₯ < 𝑦 β†’ (π‘—β€˜π‘₯) < (π‘—β€˜π‘¦))))
1541532ralbidv 3217 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = 𝑗 β†’ (βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘“β€˜π‘₯) < (π‘“β€˜π‘¦)) ↔ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘—β€˜π‘₯) < (π‘—β€˜π‘¦))))
155149, 154anbi12d 630 . . . . . . . . . . . . . . . . . . 19 (𝑓 = 𝑗 β†’ ((𝑓:(1...𝐾)⟢(1...𝑁) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘“β€˜π‘₯) < (π‘“β€˜π‘¦))) ↔ (𝑗:(1...𝐾)⟢(1...𝑁) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘—β€˜π‘₯) < (π‘—β€˜π‘¦)))))
156148, 155bibi12d 345 . . . . . . . . . . . . . . . . . 18 (𝑓 = 𝑗 β†’ ((𝑓 ∈ 𝐴 ↔ (𝑓:(1...𝐾)⟢(1...𝑁) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘“β€˜π‘₯) < (π‘“β€˜π‘¦)))) ↔ (𝑗 ∈ 𝐴 ↔ (𝑗:(1...𝐾)⟢(1...𝑁) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘—β€˜π‘₯) < (π‘—β€˜π‘¦))))))
157156, 16chvarvv 2001 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ 𝐴 ↔ (𝑗:(1...𝐾)⟢(1...𝑁) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘—β€˜π‘₯) < (π‘—β€˜π‘¦))))
158157biimpi 215 . . . . . . . . . . . . . . . 16 (𝑗 ∈ 𝐴 β†’ (𝑗:(1...𝐾)⟢(1...𝑁) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘—β€˜π‘₯) < (π‘—β€˜π‘¦))))
159158adantl 481 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝑗 ∈ 𝐴) β†’ (𝑗:(1...𝐾)⟢(1...𝑁) ∧ βˆ€π‘₯ ∈ (1...𝐾)βˆ€π‘¦ ∈ (1...𝐾)(π‘₯ < 𝑦 β†’ (π‘—β€˜π‘₯) < (π‘—β€˜π‘¦))))
160159simpld 494 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑗 ∈ 𝐴) β†’ 𝑗:(1...𝐾)⟢(1...𝑁))
1611603adant2 1130 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) β†’ 𝑗:(1...𝐾)⟢(1...𝑁))
162161frnd 6725 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) β†’ ran 𝑗 βŠ† (1...𝑁))
163147, 162sselpwd 5326 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) β†’ ran 𝑗 ∈ 𝒫 (1...𝑁))
164163adantr 480 . . . . . . . . . 10 (((πœ‘ ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 β‰  𝑗) β†’ ran 𝑗 ∈ 𝒫 (1...𝑁))
165122, 145, 114, 164fvmptd 7005 . . . . . . . . 9 (((πœ‘ ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 β‰  𝑗) β†’ (πΉβ€˜π‘—) = ran 𝑗)
166121, 143, 1653netr4d 3017 . . . . . . . 8 (((πœ‘ ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) ∧ 𝑖 β‰  𝑗) β†’ (πΉβ€˜π‘–) β‰  (πΉβ€˜π‘—))
167166ex 412 . . . . . . 7 ((πœ‘ ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) β†’ (𝑖 β‰  𝑗 β†’ (πΉβ€˜π‘–) β‰  (πΉβ€˜π‘—)))
168167necon4d 2963 . . . . . 6 ((πœ‘ ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴) β†’ ((πΉβ€˜π‘–) = (πΉβ€˜π‘—) β†’ 𝑖 = 𝑗))
1691683expa 1117 . . . . 5 (((πœ‘ ∧ 𝑖 ∈ 𝐴) ∧ 𝑗 ∈ 𝐴) β†’ ((πΉβ€˜π‘–) = (πΉβ€˜π‘—) β†’ 𝑖 = 𝑗))
170169ralrimiva 3145 . . . 4 ((πœ‘ ∧ 𝑖 ∈ 𝐴) β†’ βˆ€π‘— ∈ 𝐴 ((πΉβ€˜π‘–) = (πΉβ€˜π‘—) β†’ 𝑖 = 𝑗))
171170ralrimiva 3145 . . 3 (πœ‘ β†’ βˆ€π‘– ∈ 𝐴 βˆ€π‘— ∈ 𝐴 ((πΉβ€˜π‘–) = (πΉβ€˜π‘—) β†’ 𝑖 = 𝑗))
172107, 171jca 511 . 2 (πœ‘ β†’ (𝐹:𝐴⟢𝐡 ∧ βˆ€π‘– ∈ 𝐴 βˆ€π‘— ∈ 𝐴 ((πΉβ€˜π‘–) = (πΉβ€˜π‘—) β†’ 𝑖 = 𝑗)))
173 dff13 7257 . 2 (𝐹:𝐴–1-1→𝐡 ↔ (𝐹:𝐴⟢𝐡 ∧ βˆ€π‘– ∈ 𝐴 βˆ€π‘— ∈ 𝐴 ((πΉβ€˜π‘–) = (πΉβ€˜π‘—) β†’ 𝑖 = 𝑗)))
174172, 173sylibr 233 1 (πœ‘ β†’ 𝐹:𝐴–1-1→𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844   ∧ w3a 1086  βˆ€wal 1538   = wceq 1540   ∈ wcel 2105  {cab 2708   β‰  wne 2939  βˆ€wral 3060  {crab 3431  π’« cpw 4602   class class class wbr 5148   ↦ cmpt 5231  ran crn 5677   Fn wfn 6538  βŸΆwf 6539  β€“1-1β†’wf1 6540  β€˜cfv 6543  (class class class)co 7412  Fincfn 8943  infcinf 9440  β„cr 11113  1c1 11115   < clt 11253  β„•cn 12217  β„•0cn0 12477  ...cfz 13489  β™―chash 14295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191  ax-pre-sup 11192
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-er 8707  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-sup 9441  df-inf 9442  df-card 9938  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-n0 12478  df-z 12564  df-uz 12828  df-fz 13490  df-hash 14296
This theorem is referenced by:  sticksstones3  41271  sticksstones4  41272
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