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Theorem sticksstones19 40619
Description: Extend sticks and stones to finite sets, bijective builder. (Contributed by metakunt, 23-Oct-2024.)
Hypotheses
Ref Expression
sticksstones19.1 (πœ‘ β†’ 𝑁 ∈ β„•0)
sticksstones19.2 (πœ‘ β†’ 𝐾 ∈ β„•0)
sticksstones19.3 𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...𝐾)(π‘”β€˜π‘–) = 𝑁)}
sticksstones19.4 𝐡 = {β„Ž ∣ (β„Ž:π‘†βŸΆβ„•0 ∧ Σ𝑖 ∈ 𝑆 (β„Žβ€˜π‘–) = 𝑁)}
sticksstones19.5 (πœ‘ β†’ 𝑍:(1...𝐾)–1-1-onto→𝑆)
sticksstones19.6 𝐹 = (π‘Ž ∈ 𝐴 ↦ (π‘₯ ∈ 𝑆 ↦ (π‘Žβ€˜(β—‘π‘β€˜π‘₯))))
sticksstones19.7 𝐺 = (𝑏 ∈ 𝐡 ↦ (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦))))
Assertion
Ref Expression
sticksstones19 (πœ‘ β†’ 𝐹:𝐴–1-1-onto→𝐡)
Distinct variable groups:   𝐴,π‘Ž,𝑖,π‘₯,𝑦   𝐴,𝑏,𝑖,π‘₯,𝑦   𝐡,π‘Ž,𝑖,π‘₯,𝑦   𝐡,𝑏   𝐹,𝑏,𝑦   𝐺,π‘Ž,π‘₯   𝐾,π‘Ž,𝑔,𝑖,𝑦   𝐾,𝑏,𝑔   π‘₯,𝐾   𝑔,𝑁   β„Ž,𝑁   𝑆,π‘Ž,β„Ž,𝑖,π‘₯   𝑆,𝑏,β„Ž   𝑦,𝑆   𝑍,π‘Ž,𝑔,𝑖,𝑦   𝑍,𝑏,β„Ž,π‘₯   πœ‘,π‘Ž,𝑖,π‘₯,𝑦   πœ‘,𝑏
Allowed substitution hints:   πœ‘(𝑔,β„Ž)   𝐴(𝑔,β„Ž)   𝐡(𝑔,β„Ž)   𝑆(𝑔)   𝐹(π‘₯,𝑔,β„Ž,𝑖,π‘Ž)   𝐺(𝑦,𝑔,β„Ž,𝑖,𝑏)   𝐾(β„Ž)   𝑁(π‘₯,𝑦,𝑖,π‘Ž,𝑏)

Proof of Theorem sticksstones19
Dummy variables 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sticksstones19.1 . . 3 (πœ‘ β†’ 𝑁 ∈ β„•0)
2 sticksstones19.2 . . 3 (πœ‘ β†’ 𝐾 ∈ β„•0)
3 sticksstones19.3 . . 3 𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...𝐾)(π‘”β€˜π‘–) = 𝑁)}
4 sticksstones19.4 . . 3 𝐡 = {β„Ž ∣ (β„Ž:π‘†βŸΆβ„•0 ∧ Σ𝑖 ∈ 𝑆 (β„Žβ€˜π‘–) = 𝑁)}
5 sticksstones19.5 . . 3 (πœ‘ β†’ 𝑍:(1...𝐾)–1-1-onto→𝑆)
6 sticksstones19.6 . . 3 𝐹 = (π‘Ž ∈ 𝐴 ↦ (π‘₯ ∈ 𝑆 ↦ (π‘Žβ€˜(β—‘π‘β€˜π‘₯))))
71, 2, 3, 4, 5, 6sticksstones18 40618 . 2 (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
8 sticksstones19.7 . . 3 𝐺 = (𝑏 ∈ 𝐡 ↦ (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦))))
91, 2, 3, 4, 5, 8sticksstones17 40617 . 2 (πœ‘ β†’ 𝐺:𝐡⟢𝐴)
108a1i 11 . . . . 5 ((πœ‘ ∧ 𝑐 ∈ 𝐴) β†’ 𝐺 = (𝑏 ∈ 𝐡 ↦ (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦)))))
11 simplr 768 . . . . . . 7 ((((πœ‘ ∧ 𝑐 ∈ 𝐴) ∧ 𝑏 = (πΉβ€˜π‘)) ∧ 𝑦 ∈ (1...𝐾)) β†’ 𝑏 = (πΉβ€˜π‘))
1211fveq1d 6845 . . . . . 6 ((((πœ‘ ∧ 𝑐 ∈ 𝐴) ∧ 𝑏 = (πΉβ€˜π‘)) ∧ 𝑦 ∈ (1...𝐾)) β†’ (π‘β€˜(π‘β€˜π‘¦)) = ((πΉβ€˜π‘)β€˜(π‘β€˜π‘¦)))
1312mpteq2dva 5206 . . . . 5 (((πœ‘ ∧ 𝑐 ∈ 𝐴) ∧ 𝑏 = (πΉβ€˜π‘)) β†’ (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦))) = (𝑦 ∈ (1...𝐾) ↦ ((πΉβ€˜π‘)β€˜(π‘β€˜π‘¦))))
147ffvelcdmda 7036 . . . . 5 ((πœ‘ ∧ 𝑐 ∈ 𝐴) β†’ (πΉβ€˜π‘) ∈ 𝐡)
15 fzfid 13884 . . . . . 6 ((πœ‘ ∧ 𝑐 ∈ 𝐴) β†’ (1...𝐾) ∈ Fin)
1615mptexd 7175 . . . . 5 ((πœ‘ ∧ 𝑐 ∈ 𝐴) β†’ (𝑦 ∈ (1...𝐾) ↦ ((πΉβ€˜π‘)β€˜(π‘β€˜π‘¦))) ∈ V)
1710, 13, 14, 16fvmptd 6956 . . . 4 ((πœ‘ ∧ 𝑐 ∈ 𝐴) β†’ (πΊβ€˜(πΉβ€˜π‘)) = (𝑦 ∈ (1...𝐾) ↦ ((πΉβ€˜π‘)β€˜(π‘β€˜π‘¦))))
186a1i 11 . . . . . . . . 9 ((πœ‘ ∧ 𝑐 ∈ 𝐴 ∧ 𝑦 ∈ (1...𝐾)) β†’ 𝐹 = (π‘Ž ∈ 𝐴 ↦ (π‘₯ ∈ 𝑆 ↦ (π‘Žβ€˜(β—‘π‘β€˜π‘₯)))))
1918fveq1d 6845 . . . . . . . 8 ((πœ‘ ∧ 𝑐 ∈ 𝐴 ∧ 𝑦 ∈ (1...𝐾)) β†’ (πΉβ€˜π‘) = ((π‘Ž ∈ 𝐴 ↦ (π‘₯ ∈ 𝑆 ↦ (π‘Žβ€˜(β—‘π‘β€˜π‘₯))))β€˜π‘))
2019fveq1d 6845 . . . . . . 7 ((πœ‘ ∧ 𝑐 ∈ 𝐴 ∧ 𝑦 ∈ (1...𝐾)) β†’ ((πΉβ€˜π‘)β€˜(π‘β€˜π‘¦)) = (((π‘Ž ∈ 𝐴 ↦ (π‘₯ ∈ 𝑆 ↦ (π‘Žβ€˜(β—‘π‘β€˜π‘₯))))β€˜π‘)β€˜(π‘β€˜π‘¦)))
21203expa 1119 . . . . . 6 (((πœ‘ ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) β†’ ((πΉβ€˜π‘)β€˜(π‘β€˜π‘¦)) = (((π‘Ž ∈ 𝐴 ↦ (π‘₯ ∈ 𝑆 ↦ (π‘Žβ€˜(β—‘π‘β€˜π‘₯))))β€˜π‘)β€˜(π‘β€˜π‘¦)))
2221mpteq2dva 5206 . . . . 5 ((πœ‘ ∧ 𝑐 ∈ 𝐴) β†’ (𝑦 ∈ (1...𝐾) ↦ ((πΉβ€˜π‘)β€˜(π‘β€˜π‘¦))) = (𝑦 ∈ (1...𝐾) ↦ (((π‘Ž ∈ 𝐴 ↦ (π‘₯ ∈ 𝑆 ↦ (π‘Žβ€˜(β—‘π‘β€˜π‘₯))))β€˜π‘)β€˜(π‘β€˜π‘¦))))
23 eqidd 2734 . . . . . . . . 9 (((πœ‘ ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) β†’ (π‘Ž ∈ 𝐴 ↦ (π‘₯ ∈ 𝑆 ↦ (π‘Žβ€˜(β—‘π‘β€˜π‘₯)))) = (π‘Ž ∈ 𝐴 ↦ (π‘₯ ∈ 𝑆 ↦ (π‘Žβ€˜(β—‘π‘β€˜π‘₯)))))
24 simplr 768 . . . . . . . . . . 11 (((((πœ‘ ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) ∧ π‘Ž = 𝑐) ∧ π‘₯ ∈ 𝑆) β†’ π‘Ž = 𝑐)
2524fveq1d 6845 . . . . . . . . . 10 (((((πœ‘ ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) ∧ π‘Ž = 𝑐) ∧ π‘₯ ∈ 𝑆) β†’ (π‘Žβ€˜(β—‘π‘β€˜π‘₯)) = (π‘β€˜(β—‘π‘β€˜π‘₯)))
2625mpteq2dva 5206 . . . . . . . . 9 ((((πœ‘ ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) ∧ π‘Ž = 𝑐) β†’ (π‘₯ ∈ 𝑆 ↦ (π‘Žβ€˜(β—‘π‘β€˜π‘₯))) = (π‘₯ ∈ 𝑆 ↦ (π‘β€˜(β—‘π‘β€˜π‘₯))))
27 simplr 768 . . . . . . . . 9 (((πœ‘ ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) β†’ 𝑐 ∈ 𝐴)
28 fzfid 13884 . . . . . . . . . . . . 13 (πœ‘ β†’ (1...𝐾) ∈ Fin)
29 f1oenfi 9129 . . . . . . . . . . . . . . 15 (((1...𝐾) ∈ Fin ∧ 𝑍:(1...𝐾)–1-1-onto→𝑆) β†’ (1...𝐾) β‰ˆ 𝑆)
3028, 5, 29syl2anc 585 . . . . . . . . . . . . . 14 (πœ‘ β†’ (1...𝐾) β‰ˆ 𝑆)
3130ensymd 8948 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝑆 β‰ˆ (1...𝐾))
32 enfii 9136 . . . . . . . . . . . . 13 (((1...𝐾) ∈ Fin ∧ 𝑆 β‰ˆ (1...𝐾)) β†’ 𝑆 ∈ Fin)
3328, 31, 32syl2anc 585 . . . . . . . . . . . 12 (πœ‘ β†’ 𝑆 ∈ Fin)
3433adantr 482 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑐 ∈ 𝐴) β†’ 𝑆 ∈ Fin)
3534adantr 482 . . . . . . . . . 10 (((πœ‘ ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) β†’ 𝑆 ∈ Fin)
3635mptexd 7175 . . . . . . . . 9 (((πœ‘ ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) β†’ (π‘₯ ∈ 𝑆 ↦ (π‘β€˜(β—‘π‘β€˜π‘₯))) ∈ V)
3723, 26, 27, 36fvmptd 6956 . . . . . . . 8 (((πœ‘ ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) β†’ ((π‘Ž ∈ 𝐴 ↦ (π‘₯ ∈ 𝑆 ↦ (π‘Žβ€˜(β—‘π‘β€˜π‘₯))))β€˜π‘) = (π‘₯ ∈ 𝑆 ↦ (π‘β€˜(β—‘π‘β€˜π‘₯))))
3837fveq1d 6845 . . . . . . 7 (((πœ‘ ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) β†’ (((π‘Ž ∈ 𝐴 ↦ (π‘₯ ∈ 𝑆 ↦ (π‘Žβ€˜(β—‘π‘β€˜π‘₯))))β€˜π‘)β€˜(π‘β€˜π‘¦)) = ((π‘₯ ∈ 𝑆 ↦ (π‘β€˜(β—‘π‘β€˜π‘₯)))β€˜(π‘β€˜π‘¦)))
3938mpteq2dva 5206 . . . . . 6 ((πœ‘ ∧ 𝑐 ∈ 𝐴) β†’ (𝑦 ∈ (1...𝐾) ↦ (((π‘Ž ∈ 𝐴 ↦ (π‘₯ ∈ 𝑆 ↦ (π‘Žβ€˜(β—‘π‘β€˜π‘₯))))β€˜π‘)β€˜(π‘β€˜π‘¦))) = (𝑦 ∈ (1...𝐾) ↦ ((π‘₯ ∈ 𝑆 ↦ (π‘β€˜(β—‘π‘β€˜π‘₯)))β€˜(π‘β€˜π‘¦))))
40 eqidd 2734 . . . . . . . . 9 (((πœ‘ ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) β†’ (π‘₯ ∈ 𝑆 ↦ (π‘β€˜(β—‘π‘β€˜π‘₯))) = (π‘₯ ∈ 𝑆 ↦ (π‘β€˜(β—‘π‘β€˜π‘₯))))
41 simpr 486 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) ∧ π‘₯ = (π‘β€˜π‘¦)) β†’ π‘₯ = (π‘β€˜π‘¦))
4241fveq2d 6847 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) ∧ π‘₯ = (π‘β€˜π‘¦)) β†’ (β—‘π‘β€˜π‘₯) = (β—‘π‘β€˜(π‘β€˜π‘¦)))
4342fveq2d 6847 . . . . . . . . 9 ((((πœ‘ ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) ∧ π‘₯ = (π‘β€˜π‘¦)) β†’ (π‘β€˜(β—‘π‘β€˜π‘₯)) = (π‘β€˜(β—‘π‘β€˜(π‘β€˜π‘¦))))
44 f1of 6785 . . . . . . . . . . . 12 (𝑍:(1...𝐾)–1-1-onto→𝑆 β†’ 𝑍:(1...𝐾)βŸΆπ‘†)
455, 44syl 17 . . . . . . . . . . 11 (πœ‘ β†’ 𝑍:(1...𝐾)βŸΆπ‘†)
4645adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ 𝑐 ∈ 𝐴) β†’ 𝑍:(1...𝐾)βŸΆπ‘†)
4746ffvelcdmda 7036 . . . . . . . . 9 (((πœ‘ ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) β†’ (π‘β€˜π‘¦) ∈ 𝑆)
48 fvexd 6858 . . . . . . . . 9 (((πœ‘ ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) β†’ (π‘β€˜(β—‘π‘β€˜(π‘β€˜π‘¦))) ∈ V)
4940, 43, 47, 48fvmptd 6956 . . . . . . . 8 (((πœ‘ ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) β†’ ((π‘₯ ∈ 𝑆 ↦ (π‘β€˜(β—‘π‘β€˜π‘₯)))β€˜(π‘β€˜π‘¦)) = (π‘β€˜(β—‘π‘β€˜(π‘β€˜π‘¦))))
5049mpteq2dva 5206 . . . . . . 7 ((πœ‘ ∧ 𝑐 ∈ 𝐴) β†’ (𝑦 ∈ (1...𝐾) ↦ ((π‘₯ ∈ 𝑆 ↦ (π‘β€˜(β—‘π‘β€˜π‘₯)))β€˜(π‘β€˜π‘¦))) = (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(β—‘π‘β€˜(π‘β€˜π‘¦)))))
515ad2antrr 725 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) β†’ 𝑍:(1...𝐾)–1-1-onto→𝑆)
52 simpr 486 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) β†’ 𝑦 ∈ (1...𝐾))
53 f1ocnvfv1 7223 . . . . . . . . . . 11 ((𝑍:(1...𝐾)–1-1-onto→𝑆 ∧ 𝑦 ∈ (1...𝐾)) β†’ (β—‘π‘β€˜(π‘β€˜π‘¦)) = 𝑦)
5451, 52, 53syl2anc 585 . . . . . . . . . 10 (((πœ‘ ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) β†’ (β—‘π‘β€˜(π‘β€˜π‘¦)) = 𝑦)
5554fveq2d 6847 . . . . . . . . 9 (((πœ‘ ∧ 𝑐 ∈ 𝐴) ∧ 𝑦 ∈ (1...𝐾)) β†’ (π‘β€˜(β—‘π‘β€˜(π‘β€˜π‘¦))) = (π‘β€˜π‘¦))
5655mpteq2dva 5206 . . . . . . . 8 ((πœ‘ ∧ 𝑐 ∈ 𝐴) β†’ (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(β—‘π‘β€˜(π‘β€˜π‘¦)))) = (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜π‘¦)))
57 simpr 486 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑐 ∈ 𝐴) β†’ 𝑐 ∈ 𝐴)
583a1i 11 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑐 ∈ 𝐴) β†’ 𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...𝐾)(π‘”β€˜π‘–) = 𝑁)})
5957, 58eleqtrd 2836 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑐 ∈ 𝐴) β†’ 𝑐 ∈ {𝑔 ∣ (𝑔:(1...𝐾)βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...𝐾)(π‘”β€˜π‘–) = 𝑁)})
60 vex 3448 . . . . . . . . . . . . . 14 𝑐 ∈ V
61 feq1 6650 . . . . . . . . . . . . . . 15 (𝑔 = 𝑐 β†’ (𝑔:(1...𝐾)βŸΆβ„•0 ↔ 𝑐:(1...𝐾)βŸΆβ„•0))
62 simpl 484 . . . . . . . . . . . . . . . . . 18 ((𝑔 = 𝑐 ∧ 𝑖 ∈ (1...𝐾)) β†’ 𝑔 = 𝑐)
6362fveq1d 6845 . . . . . . . . . . . . . . . . 17 ((𝑔 = 𝑐 ∧ 𝑖 ∈ (1...𝐾)) β†’ (π‘”β€˜π‘–) = (π‘β€˜π‘–))
6463sumeq2dv 15593 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑐 β†’ Σ𝑖 ∈ (1...𝐾)(π‘”β€˜π‘–) = Σ𝑖 ∈ (1...𝐾)(π‘β€˜π‘–))
6564eqeq1d 2735 . . . . . . . . . . . . . . 15 (𝑔 = 𝑐 β†’ (Σ𝑖 ∈ (1...𝐾)(π‘”β€˜π‘–) = 𝑁 ↔ Σ𝑖 ∈ (1...𝐾)(π‘β€˜π‘–) = 𝑁))
6661, 65anbi12d 632 . . . . . . . . . . . . . 14 (𝑔 = 𝑐 β†’ ((𝑔:(1...𝐾)βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...𝐾)(π‘”β€˜π‘–) = 𝑁) ↔ (𝑐:(1...𝐾)βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...𝐾)(π‘β€˜π‘–) = 𝑁)))
6760, 66elab 3631 . . . . . . . . . . . . 13 (𝑐 ∈ {𝑔 ∣ (𝑔:(1...𝐾)βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...𝐾)(π‘”β€˜π‘–) = 𝑁)} ↔ (𝑐:(1...𝐾)βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...𝐾)(π‘β€˜π‘–) = 𝑁))
6859, 67sylib 217 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑐 ∈ 𝐴) β†’ (𝑐:(1...𝐾)βŸΆβ„•0 ∧ Σ𝑖 ∈ (1...𝐾)(π‘β€˜π‘–) = 𝑁))
6968simpld 496 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑐 ∈ 𝐴) β†’ 𝑐:(1...𝐾)βŸΆβ„•0)
70 ffn 6669 . . . . . . . . . . 11 (𝑐:(1...𝐾)βŸΆβ„•0 β†’ 𝑐 Fn (1...𝐾))
7169, 70syl 17 . . . . . . . . . 10 ((πœ‘ ∧ 𝑐 ∈ 𝐴) β†’ 𝑐 Fn (1...𝐾))
72 dffn5 6902 . . . . . . . . . 10 (𝑐 Fn (1...𝐾) ↔ 𝑐 = (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜π‘¦)))
7371, 72sylib 217 . . . . . . . . 9 ((πœ‘ ∧ 𝑐 ∈ 𝐴) β†’ 𝑐 = (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜π‘¦)))
7473eqcomd 2739 . . . . . . . 8 ((πœ‘ ∧ 𝑐 ∈ 𝐴) β†’ (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜π‘¦)) = 𝑐)
7556, 74eqtrd 2773 . . . . . . 7 ((πœ‘ ∧ 𝑐 ∈ 𝐴) β†’ (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(β—‘π‘β€˜(π‘β€˜π‘¦)))) = 𝑐)
7650, 75eqtrd 2773 . . . . . 6 ((πœ‘ ∧ 𝑐 ∈ 𝐴) β†’ (𝑦 ∈ (1...𝐾) ↦ ((π‘₯ ∈ 𝑆 ↦ (π‘β€˜(β—‘π‘β€˜π‘₯)))β€˜(π‘β€˜π‘¦))) = 𝑐)
7739, 76eqtrd 2773 . . . . 5 ((πœ‘ ∧ 𝑐 ∈ 𝐴) β†’ (𝑦 ∈ (1...𝐾) ↦ (((π‘Ž ∈ 𝐴 ↦ (π‘₯ ∈ 𝑆 ↦ (π‘Žβ€˜(β—‘π‘β€˜π‘₯))))β€˜π‘)β€˜(π‘β€˜π‘¦))) = 𝑐)
7822, 77eqtrd 2773 . . . 4 ((πœ‘ ∧ 𝑐 ∈ 𝐴) β†’ (𝑦 ∈ (1...𝐾) ↦ ((πΉβ€˜π‘)β€˜(π‘β€˜π‘¦))) = 𝑐)
7917, 78eqtrd 2773 . . 3 ((πœ‘ ∧ 𝑐 ∈ 𝐴) β†’ (πΊβ€˜(πΉβ€˜π‘)) = 𝑐)
8079ralrimiva 3140 . 2 (πœ‘ β†’ βˆ€π‘ ∈ 𝐴 (πΊβ€˜(πΉβ€˜π‘)) = 𝑐)
816a1i 11 . . . . 5 ((πœ‘ ∧ 𝑑 ∈ 𝐡) β†’ 𝐹 = (π‘Ž ∈ 𝐴 ↦ (π‘₯ ∈ 𝑆 ↦ (π‘Žβ€˜(β—‘π‘β€˜π‘₯)))))
82 simplr 768 . . . . . . 7 ((((πœ‘ ∧ 𝑑 ∈ 𝐡) ∧ π‘Ž = (πΊβ€˜π‘‘)) ∧ π‘₯ ∈ 𝑆) β†’ π‘Ž = (πΊβ€˜π‘‘))
8382fveq1d 6845 . . . . . 6 ((((πœ‘ ∧ 𝑑 ∈ 𝐡) ∧ π‘Ž = (πΊβ€˜π‘‘)) ∧ π‘₯ ∈ 𝑆) β†’ (π‘Žβ€˜(β—‘π‘β€˜π‘₯)) = ((πΊβ€˜π‘‘)β€˜(β—‘π‘β€˜π‘₯)))
8483mpteq2dva 5206 . . . . 5 (((πœ‘ ∧ 𝑑 ∈ 𝐡) ∧ π‘Ž = (πΊβ€˜π‘‘)) β†’ (π‘₯ ∈ 𝑆 ↦ (π‘Žβ€˜(β—‘π‘β€˜π‘₯))) = (π‘₯ ∈ 𝑆 ↦ ((πΊβ€˜π‘‘)β€˜(β—‘π‘β€˜π‘₯))))
859ffvelcdmda 7036 . . . . 5 ((πœ‘ ∧ 𝑑 ∈ 𝐡) β†’ (πΊβ€˜π‘‘) ∈ 𝐴)
8633adantr 482 . . . . . 6 ((πœ‘ ∧ 𝑑 ∈ 𝐡) β†’ 𝑆 ∈ Fin)
8786mptexd 7175 . . . . 5 ((πœ‘ ∧ 𝑑 ∈ 𝐡) β†’ (π‘₯ ∈ 𝑆 ↦ ((πΊβ€˜π‘‘)β€˜(β—‘π‘β€˜π‘₯))) ∈ V)
8881, 84, 85, 87fvmptd 6956 . . . 4 ((πœ‘ ∧ 𝑑 ∈ 𝐡) β†’ (πΉβ€˜(πΊβ€˜π‘‘)) = (π‘₯ ∈ 𝑆 ↦ ((πΊβ€˜π‘‘)β€˜(β—‘π‘β€˜π‘₯))))
898a1i 11 . . . . . . . 8 (((πœ‘ ∧ 𝑑 ∈ 𝐡) ∧ π‘₯ ∈ 𝑆) β†’ 𝐺 = (𝑏 ∈ 𝐡 ↦ (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦)))))
90 simplr 768 . . . . . . . . . 10 (((((πœ‘ ∧ 𝑑 ∈ 𝐡) ∧ π‘₯ ∈ 𝑆) ∧ 𝑏 = 𝑑) ∧ 𝑦 ∈ (1...𝐾)) β†’ 𝑏 = 𝑑)
9190fveq1d 6845 . . . . . . . . 9 (((((πœ‘ ∧ 𝑑 ∈ 𝐡) ∧ π‘₯ ∈ 𝑆) ∧ 𝑏 = 𝑑) ∧ 𝑦 ∈ (1...𝐾)) β†’ (π‘β€˜(π‘β€˜π‘¦)) = (π‘‘β€˜(π‘β€˜π‘¦)))
9291mpteq2dva 5206 . . . . . . . 8 ((((πœ‘ ∧ 𝑑 ∈ 𝐡) ∧ π‘₯ ∈ 𝑆) ∧ 𝑏 = 𝑑) β†’ (𝑦 ∈ (1...𝐾) ↦ (π‘β€˜(π‘β€˜π‘¦))) = (𝑦 ∈ (1...𝐾) ↦ (π‘‘β€˜(π‘β€˜π‘¦))))
93 simplr 768 . . . . . . . 8 (((πœ‘ ∧ 𝑑 ∈ 𝐡) ∧ π‘₯ ∈ 𝑆) β†’ 𝑑 ∈ 𝐡)
94 fzfid 13884 . . . . . . . . 9 (((πœ‘ ∧ 𝑑 ∈ 𝐡) ∧ π‘₯ ∈ 𝑆) β†’ (1...𝐾) ∈ Fin)
9594mptexd 7175 . . . . . . . 8 (((πœ‘ ∧ 𝑑 ∈ 𝐡) ∧ π‘₯ ∈ 𝑆) β†’ (𝑦 ∈ (1...𝐾) ↦ (π‘‘β€˜(π‘β€˜π‘¦))) ∈ V)
9689, 92, 93, 95fvmptd 6956 . . . . . . 7 (((πœ‘ ∧ 𝑑 ∈ 𝐡) ∧ π‘₯ ∈ 𝑆) β†’ (πΊβ€˜π‘‘) = (𝑦 ∈ (1...𝐾) ↦ (π‘‘β€˜(π‘β€˜π‘¦))))
9796fveq1d 6845 . . . . . 6 (((πœ‘ ∧ 𝑑 ∈ 𝐡) ∧ π‘₯ ∈ 𝑆) β†’ ((πΊβ€˜π‘‘)β€˜(β—‘π‘β€˜π‘₯)) = ((𝑦 ∈ (1...𝐾) ↦ (π‘‘β€˜(π‘β€˜π‘¦)))β€˜(β—‘π‘β€˜π‘₯)))
9897mpteq2dva 5206 . . . . 5 ((πœ‘ ∧ 𝑑 ∈ 𝐡) β†’ (π‘₯ ∈ 𝑆 ↦ ((πΊβ€˜π‘‘)β€˜(β—‘π‘β€˜π‘₯))) = (π‘₯ ∈ 𝑆 ↦ ((𝑦 ∈ (1...𝐾) ↦ (π‘‘β€˜(π‘β€˜π‘¦)))β€˜(β—‘π‘β€˜π‘₯))))
99 eqidd 2734 . . . . . . . 8 (((πœ‘ ∧ 𝑑 ∈ 𝐡) ∧ π‘₯ ∈ 𝑆) β†’ (𝑦 ∈ (1...𝐾) ↦ (π‘‘β€˜(π‘β€˜π‘¦))) = (𝑦 ∈ (1...𝐾) ↦ (π‘‘β€˜(π‘β€˜π‘¦))))
100 simpr 486 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑑 ∈ 𝐡) ∧ π‘₯ ∈ 𝑆) ∧ 𝑦 = (β—‘π‘β€˜π‘₯)) β†’ 𝑦 = (β—‘π‘β€˜π‘₯))
101100fveq2d 6847 . . . . . . . . 9 ((((πœ‘ ∧ 𝑑 ∈ 𝐡) ∧ π‘₯ ∈ 𝑆) ∧ 𝑦 = (β—‘π‘β€˜π‘₯)) β†’ (π‘β€˜π‘¦) = (π‘β€˜(β—‘π‘β€˜π‘₯)))
102101fveq2d 6847 . . . . . . . 8 ((((πœ‘ ∧ 𝑑 ∈ 𝐡) ∧ π‘₯ ∈ 𝑆) ∧ 𝑦 = (β—‘π‘β€˜π‘₯)) β†’ (π‘‘β€˜(π‘β€˜π‘¦)) = (π‘‘β€˜(π‘β€˜(β—‘π‘β€˜π‘₯))))
103 f1ocnv 6797 . . . . . . . . . . . 12 (𝑍:(1...𝐾)–1-1-onto→𝑆 β†’ ◑𝑍:𝑆–1-1-ontoβ†’(1...𝐾))
1045, 103syl 17 . . . . . . . . . . 11 (πœ‘ β†’ ◑𝑍:𝑆–1-1-ontoβ†’(1...𝐾))
105 f1of 6785 . . . . . . . . . . 11 (◑𝑍:𝑆–1-1-ontoβ†’(1...𝐾) β†’ ◑𝑍:π‘†βŸΆ(1...𝐾))
106104, 105syl 17 . . . . . . . . . 10 (πœ‘ β†’ ◑𝑍:π‘†βŸΆ(1...𝐾))
107106adantr 482 . . . . . . . . 9 ((πœ‘ ∧ 𝑑 ∈ 𝐡) β†’ ◑𝑍:π‘†βŸΆ(1...𝐾))
108107ffvelcdmda 7036 . . . . . . . 8 (((πœ‘ ∧ 𝑑 ∈ 𝐡) ∧ π‘₯ ∈ 𝑆) β†’ (β—‘π‘β€˜π‘₯) ∈ (1...𝐾))
109 fvexd 6858 . . . . . . . 8 (((πœ‘ ∧ 𝑑 ∈ 𝐡) ∧ π‘₯ ∈ 𝑆) β†’ (π‘‘β€˜(π‘β€˜(β—‘π‘β€˜π‘₯))) ∈ V)
11099, 102, 108, 109fvmptd 6956 . . . . . . 7 (((πœ‘ ∧ 𝑑 ∈ 𝐡) ∧ π‘₯ ∈ 𝑆) β†’ ((𝑦 ∈ (1...𝐾) ↦ (π‘‘β€˜(π‘β€˜π‘¦)))β€˜(β—‘π‘β€˜π‘₯)) = (π‘‘β€˜(π‘β€˜(β—‘π‘β€˜π‘₯))))
111110mpteq2dva 5206 . . . . . 6 ((πœ‘ ∧ 𝑑 ∈ 𝐡) β†’ (π‘₯ ∈ 𝑆 ↦ ((𝑦 ∈ (1...𝐾) ↦ (π‘‘β€˜(π‘β€˜π‘¦)))β€˜(β—‘π‘β€˜π‘₯))) = (π‘₯ ∈ 𝑆 ↦ (π‘‘β€˜(π‘β€˜(β—‘π‘β€˜π‘₯)))))
1125ad2antrr 725 . . . . . . . . . 10 (((πœ‘ ∧ 𝑑 ∈ 𝐡) ∧ π‘₯ ∈ 𝑆) β†’ 𝑍:(1...𝐾)–1-1-onto→𝑆)
113 simpr 486 . . . . . . . . . 10 (((πœ‘ ∧ 𝑑 ∈ 𝐡) ∧ π‘₯ ∈ 𝑆) β†’ π‘₯ ∈ 𝑆)
114 f1ocnvfv2 7224 . . . . . . . . . 10 ((𝑍:(1...𝐾)–1-1-onto→𝑆 ∧ π‘₯ ∈ 𝑆) β†’ (π‘β€˜(β—‘π‘β€˜π‘₯)) = π‘₯)
115112, 113, 114syl2anc 585 . . . . . . . . 9 (((πœ‘ ∧ 𝑑 ∈ 𝐡) ∧ π‘₯ ∈ 𝑆) β†’ (π‘β€˜(β—‘π‘β€˜π‘₯)) = π‘₯)
116115fveq2d 6847 . . . . . . . 8 (((πœ‘ ∧ 𝑑 ∈ 𝐡) ∧ π‘₯ ∈ 𝑆) β†’ (π‘‘β€˜(π‘β€˜(β—‘π‘β€˜π‘₯))) = (π‘‘β€˜π‘₯))
117116mpteq2dva 5206 . . . . . . 7 ((πœ‘ ∧ 𝑑 ∈ 𝐡) β†’ (π‘₯ ∈ 𝑆 ↦ (π‘‘β€˜(π‘β€˜(β—‘π‘β€˜π‘₯)))) = (π‘₯ ∈ 𝑆 ↦ (π‘‘β€˜π‘₯)))
118 simpr 486 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑑 ∈ 𝐡) β†’ 𝑑 ∈ 𝐡)
1194a1i 11 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑑 ∈ 𝐡) β†’ 𝐡 = {β„Ž ∣ (β„Ž:π‘†βŸΆβ„•0 ∧ Σ𝑖 ∈ 𝑆 (β„Žβ€˜π‘–) = 𝑁)})
120118, 119eleqtrd 2836 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑑 ∈ 𝐡) β†’ 𝑑 ∈ {β„Ž ∣ (β„Ž:π‘†βŸΆβ„•0 ∧ Σ𝑖 ∈ 𝑆 (β„Žβ€˜π‘–) = 𝑁)})
121 vex 3448 . . . . . . . . . . . . 13 𝑑 ∈ V
122 feq1 6650 . . . . . . . . . . . . . 14 (β„Ž = 𝑑 β†’ (β„Ž:π‘†βŸΆβ„•0 ↔ 𝑑:π‘†βŸΆβ„•0))
123 simpl 484 . . . . . . . . . . . . . . . . 17 ((β„Ž = 𝑑 ∧ 𝑖 ∈ 𝑆) β†’ β„Ž = 𝑑)
124123fveq1d 6845 . . . . . . . . . . . . . . . 16 ((β„Ž = 𝑑 ∧ 𝑖 ∈ 𝑆) β†’ (β„Žβ€˜π‘–) = (π‘‘β€˜π‘–))
125124sumeq2dv 15593 . . . . . . . . . . . . . . 15 (β„Ž = 𝑑 β†’ Σ𝑖 ∈ 𝑆 (β„Žβ€˜π‘–) = Σ𝑖 ∈ 𝑆 (π‘‘β€˜π‘–))
126125eqeq1d 2735 . . . . . . . . . . . . . 14 (β„Ž = 𝑑 β†’ (Σ𝑖 ∈ 𝑆 (β„Žβ€˜π‘–) = 𝑁 ↔ Σ𝑖 ∈ 𝑆 (π‘‘β€˜π‘–) = 𝑁))
127122, 126anbi12d 632 . . . . . . . . . . . . 13 (β„Ž = 𝑑 β†’ ((β„Ž:π‘†βŸΆβ„•0 ∧ Σ𝑖 ∈ 𝑆 (β„Žβ€˜π‘–) = 𝑁) ↔ (𝑑:π‘†βŸΆβ„•0 ∧ Σ𝑖 ∈ 𝑆 (π‘‘β€˜π‘–) = 𝑁)))
128121, 127elab 3631 . . . . . . . . . . . 12 (𝑑 ∈ {β„Ž ∣ (β„Ž:π‘†βŸΆβ„•0 ∧ Σ𝑖 ∈ 𝑆 (β„Žβ€˜π‘–) = 𝑁)} ↔ (𝑑:π‘†βŸΆβ„•0 ∧ Σ𝑖 ∈ 𝑆 (π‘‘β€˜π‘–) = 𝑁))
129120, 128sylib 217 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑑 ∈ 𝐡) β†’ (𝑑:π‘†βŸΆβ„•0 ∧ Σ𝑖 ∈ 𝑆 (π‘‘β€˜π‘–) = 𝑁))
130129simpld 496 . . . . . . . . . 10 ((πœ‘ ∧ 𝑑 ∈ 𝐡) β†’ 𝑑:π‘†βŸΆβ„•0)
131 ffn 6669 . . . . . . . . . 10 (𝑑:π‘†βŸΆβ„•0 β†’ 𝑑 Fn 𝑆)
132130, 131syl 17 . . . . . . . . 9 ((πœ‘ ∧ 𝑑 ∈ 𝐡) β†’ 𝑑 Fn 𝑆)
133 dffn5 6902 . . . . . . . . 9 (𝑑 Fn 𝑆 ↔ 𝑑 = (π‘₯ ∈ 𝑆 ↦ (π‘‘β€˜π‘₯)))
134132, 133sylib 217 . . . . . . . 8 ((πœ‘ ∧ 𝑑 ∈ 𝐡) β†’ 𝑑 = (π‘₯ ∈ 𝑆 ↦ (π‘‘β€˜π‘₯)))
135134eqcomd 2739 . . . . . . 7 ((πœ‘ ∧ 𝑑 ∈ 𝐡) β†’ (π‘₯ ∈ 𝑆 ↦ (π‘‘β€˜π‘₯)) = 𝑑)
136117, 135eqtrd 2773 . . . . . 6 ((πœ‘ ∧ 𝑑 ∈ 𝐡) β†’ (π‘₯ ∈ 𝑆 ↦ (π‘‘β€˜(π‘β€˜(β—‘π‘β€˜π‘₯)))) = 𝑑)
137111, 136eqtrd 2773 . . . . 5 ((πœ‘ ∧ 𝑑 ∈ 𝐡) β†’ (π‘₯ ∈ 𝑆 ↦ ((𝑦 ∈ (1...𝐾) ↦ (π‘‘β€˜(π‘β€˜π‘¦)))β€˜(β—‘π‘β€˜π‘₯))) = 𝑑)
13898, 137eqtrd 2773 . . . 4 ((πœ‘ ∧ 𝑑 ∈ 𝐡) β†’ (π‘₯ ∈ 𝑆 ↦ ((πΊβ€˜π‘‘)β€˜(β—‘π‘β€˜π‘₯))) = 𝑑)
13988, 138eqtrd 2773 . . 3 ((πœ‘ ∧ 𝑑 ∈ 𝐡) β†’ (πΉβ€˜(πΊβ€˜π‘‘)) = 𝑑)
140139ralrimiva 3140 . 2 (πœ‘ β†’ βˆ€π‘‘ ∈ 𝐡 (πΉβ€˜(πΊβ€˜π‘‘)) = 𝑑)
1417, 9, 80, 1402fvidf1od 7245 1 (πœ‘ β†’ 𝐹:𝐴–1-1-onto→𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {cab 2710  Vcvv 3444   class class class wbr 5106   ↦ cmpt 5189  β—‘ccnv 5633   Fn wfn 6492  βŸΆwf 6493  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497  (class class class)co 7358   β‰ˆ cen 8883  Fincfn 8886  1c1 11057  β„•0cn0 12418  ...cfz 13430  Ξ£csu 15576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9582  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-pre-sup 11134
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9383  df-oi 9451  df-card 9880  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-div 11818  df-nn 12159  df-2 12221  df-3 12222  df-n0 12419  df-z 12505  df-uz 12769  df-rp 12921  df-fz 13431  df-fzo 13574  df-seq 13913  df-exp 13974  df-hash 14237  df-cj 14990  df-re 14991  df-im 14992  df-sqrt 15126  df-abs 15127  df-clim 15376  df-sum 15577
This theorem is referenced by:  sticksstones20  40620
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