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Theorem pwcfsdom 10598
Description: A corollary of Konig's Theorem konigth 10584. Theorem 11.28 of [TakeutiZaring] p. 108. (Contributed by Mario Carneiro, 20-Mar-2013.)
Hypothesis
Ref Expression
pwcfsdom.1 𝐻 = (𝑦 ∈ (cfβ€˜(β„΅β€˜π΄)) ↦ (harβ€˜(π‘“β€˜π‘¦)))
Assertion
Ref Expression
pwcfsdom (β„΅β€˜π΄) β‰Ί ((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄)))
Distinct variable group:   𝐴,𝑓,𝑦
Allowed substitution hints:   𝐻(𝑦,𝑓)

Proof of Theorem pwcfsdom
Dummy variables 𝑀 𝑧 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 onzsl 7844 . . . 4 (𝐴 ∈ On ↔ (𝐴 = βˆ… ∨ βˆƒπ‘₯ ∈ On 𝐴 = suc π‘₯ ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
21biimpi 215 . . 3 (𝐴 ∈ On β†’ (𝐴 = βˆ… ∨ βˆƒπ‘₯ ∈ On 𝐴 = suc π‘₯ ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
3 cfom 10279 . . . . . . 7 (cfβ€˜Ο‰) = Ο‰
4 aleph0 10081 . . . . . . . 8 (β„΅β€˜βˆ…) = Ο‰
54fveq2i 6894 . . . . . . 7 (cfβ€˜(β„΅β€˜βˆ…)) = (cfβ€˜Ο‰)
63, 5, 43eqtr4i 2765 . . . . . 6 (cfβ€˜(β„΅β€˜βˆ…)) = (β„΅β€˜βˆ…)
7 2fveq3 6896 . . . . . 6 (𝐴 = βˆ… β†’ (cfβ€˜(β„΅β€˜π΄)) = (cfβ€˜(β„΅β€˜βˆ…)))
8 fveq2 6891 . . . . . 6 (𝐴 = βˆ… β†’ (β„΅β€˜π΄) = (β„΅β€˜βˆ…))
96, 7, 83eqtr4a 2793 . . . . 5 (𝐴 = βˆ… β†’ (cfβ€˜(β„΅β€˜π΄)) = (β„΅β€˜π΄))
10 fvex 6904 . . . . . . . . 9 (β„΅β€˜π΄) ∈ V
1110canth2 9146 . . . . . . . 8 (β„΅β€˜π΄) β‰Ί 𝒫 (β„΅β€˜π΄)
1210pw2en 9095 . . . . . . . 8 𝒫 (β„΅β€˜π΄) β‰ˆ (2o ↑m (β„΅β€˜π΄))
13 sdomentr 9127 . . . . . . . 8 (((β„΅β€˜π΄) β‰Ί 𝒫 (β„΅β€˜π΄) ∧ 𝒫 (β„΅β€˜π΄) β‰ˆ (2o ↑m (β„΅β€˜π΄))) β†’ (β„΅β€˜π΄) β‰Ί (2o ↑m (β„΅β€˜π΄)))
1411, 12, 13mp2an 691 . . . . . . 7 (β„΅β€˜π΄) β‰Ί (2o ↑m (β„΅β€˜π΄))
15 alephon 10084 . . . . . . . . 9 (β„΅β€˜π΄) ∈ On
16 alephgeom 10097 . . . . . . . . . 10 (𝐴 ∈ On ↔ Ο‰ βŠ† (β„΅β€˜π΄))
17 omelon 9661 . . . . . . . . . . . 12 Ο‰ ∈ On
18 2onn 8656 . . . . . . . . . . . 12 2o ∈ Ο‰
19 onelss 6405 . . . . . . . . . . . 12 (Ο‰ ∈ On β†’ (2o ∈ Ο‰ β†’ 2o βŠ† Ο‰))
2017, 18, 19mp2 9 . . . . . . . . . . 11 2o βŠ† Ο‰
21 sstr 3986 . . . . . . . . . . 11 ((2o βŠ† Ο‰ ∧ Ο‰ βŠ† (β„΅β€˜π΄)) β†’ 2o βŠ† (β„΅β€˜π΄))
2220, 21mpan 689 . . . . . . . . . 10 (Ο‰ βŠ† (β„΅β€˜π΄) β†’ 2o βŠ† (β„΅β€˜π΄))
2316, 22sylbi 216 . . . . . . . . 9 (𝐴 ∈ On β†’ 2o βŠ† (β„΅β€˜π΄))
24 ssdomg 9012 . . . . . . . . 9 ((β„΅β€˜π΄) ∈ On β†’ (2o βŠ† (β„΅β€˜π΄) β†’ 2o β‰Ό (β„΅β€˜π΄)))
2515, 23, 24mpsyl 68 . . . . . . . 8 (𝐴 ∈ On β†’ 2o β‰Ό (β„΅β€˜π΄))
26 mapdom1 9158 . . . . . . . 8 (2o β‰Ό (β„΅β€˜π΄) β†’ (2o ↑m (β„΅β€˜π΄)) β‰Ό ((β„΅β€˜π΄) ↑m (β„΅β€˜π΄)))
2725, 26syl 17 . . . . . . 7 (𝐴 ∈ On β†’ (2o ↑m (β„΅β€˜π΄)) β‰Ό ((β„΅β€˜π΄) ↑m (β„΅β€˜π΄)))
28 sdomdomtr 9126 . . . . . . 7 (((β„΅β€˜π΄) β‰Ί (2o ↑m (β„΅β€˜π΄)) ∧ (2o ↑m (β„΅β€˜π΄)) β‰Ό ((β„΅β€˜π΄) ↑m (β„΅β€˜π΄))) β†’ (β„΅β€˜π΄) β‰Ί ((β„΅β€˜π΄) ↑m (β„΅β€˜π΄)))
2914, 27, 28sylancr 586 . . . . . 6 (𝐴 ∈ On β†’ (β„΅β€˜π΄) β‰Ί ((β„΅β€˜π΄) ↑m (β„΅β€˜π΄)))
30 oveq2 7422 . . . . . . 7 ((cfβ€˜(β„΅β€˜π΄)) = (β„΅β€˜π΄) β†’ ((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄))) = ((β„΅β€˜π΄) ↑m (β„΅β€˜π΄)))
3130breq2d 5154 . . . . . 6 ((cfβ€˜(β„΅β€˜π΄)) = (β„΅β€˜π΄) β†’ ((β„΅β€˜π΄) β‰Ί ((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄))) ↔ (β„΅β€˜π΄) β‰Ί ((β„΅β€˜π΄) ↑m (β„΅β€˜π΄))))
3229, 31syl5ibrcom 246 . . . . 5 (𝐴 ∈ On β†’ ((cfβ€˜(β„΅β€˜π΄)) = (β„΅β€˜π΄) β†’ (β„΅β€˜π΄) β‰Ί ((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄)))))
339, 32syl5 34 . . . 4 (𝐴 ∈ On β†’ (𝐴 = βˆ… β†’ (β„΅β€˜π΄) β‰Ί ((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄)))))
34 alephreg 10597 . . . . . . 7 (cfβ€˜(β„΅β€˜suc π‘₯)) = (β„΅β€˜suc π‘₯)
35 2fveq3 6896 . . . . . . 7 (𝐴 = suc π‘₯ β†’ (cfβ€˜(β„΅β€˜π΄)) = (cfβ€˜(β„΅β€˜suc π‘₯)))
36 fveq2 6891 . . . . . . 7 (𝐴 = suc π‘₯ β†’ (β„΅β€˜π΄) = (β„΅β€˜suc π‘₯))
3734, 35, 363eqtr4a 2793 . . . . . 6 (𝐴 = suc π‘₯ β†’ (cfβ€˜(β„΅β€˜π΄)) = (β„΅β€˜π΄))
3837rexlimivw 3146 . . . . 5 (βˆƒπ‘₯ ∈ On 𝐴 = suc π‘₯ β†’ (cfβ€˜(β„΅β€˜π΄)) = (β„΅β€˜π΄))
3938, 32syl5 34 . . . 4 (𝐴 ∈ On β†’ (βˆƒπ‘₯ ∈ On 𝐴 = suc π‘₯ β†’ (β„΅β€˜π΄) β‰Ί ((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄)))))
40 cfsmo 10286 . . . . . 6 ((β„΅β€˜π΄) ∈ On β†’ βˆƒπ‘“(𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) ∧ Smo 𝑓 ∧ βˆ€π‘§ ∈ (β„΅β€˜π΄)βˆƒπ‘€ ∈ (cfβ€˜(β„΅β€˜π΄))𝑧 βŠ† (π‘“β€˜π‘€)))
41 limelon 6427 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ Lim 𝐴) β†’ 𝐴 ∈ On)
42 ffn 6716 . . . . . . . . . . . . . . . 16 (𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) β†’ 𝑓 Fn (cfβ€˜(β„΅β€˜π΄)))
43 fnrnfv 6952 . . . . . . . . . . . . . . . . 17 (𝑓 Fn (cfβ€˜(β„΅β€˜π΄)) β†’ ran 𝑓 = {𝑦 ∣ βˆƒπ‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))𝑦 = (π‘“β€˜π‘₯)})
4443unieqd 4916 . . . . . . . . . . . . . . . 16 (𝑓 Fn (cfβ€˜(β„΅β€˜π΄)) β†’ βˆͺ ran 𝑓 = βˆͺ {𝑦 ∣ βˆƒπ‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))𝑦 = (π‘“β€˜π‘₯)})
4542, 44syl 17 . . . . . . . . . . . . . . 15 (𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) β†’ βˆͺ ran 𝑓 = βˆͺ {𝑦 ∣ βˆƒπ‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))𝑦 = (π‘“β€˜π‘₯)})
46 fvex 6904 . . . . . . . . . . . . . . . 16 (π‘“β€˜π‘₯) ∈ V
4746dfiun2 5030 . . . . . . . . . . . . . . 15 βˆͺ π‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(π‘“β€˜π‘₯) = βˆͺ {𝑦 ∣ βˆƒπ‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))𝑦 = (π‘“β€˜π‘₯)}
4845, 47eqtr4di 2785 . . . . . . . . . . . . . 14 (𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) β†’ βˆͺ ran 𝑓 = βˆͺ π‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(π‘“β€˜π‘₯))
4948ad2antrl 727 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ (𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) ∧ βˆ€π‘§ ∈ (β„΅β€˜π΄)βˆƒπ‘€ ∈ (cfβ€˜(β„΅β€˜π΄))𝑧 βŠ† (π‘“β€˜π‘€))) β†’ βˆͺ ran 𝑓 = βˆͺ π‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(π‘“β€˜π‘₯))
50 fnfvelrn 7084 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 Fn (cfβ€˜(β„΅β€˜π΄)) ∧ 𝑀 ∈ (cfβ€˜(β„΅β€˜π΄))) β†’ (π‘“β€˜π‘€) ∈ ran 𝑓)
5142, 50sylan 579 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) ∧ 𝑀 ∈ (cfβ€˜(β„΅β€˜π΄))) β†’ (π‘“β€˜π‘€) ∈ ran 𝑓)
52 sseq2 4004 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (π‘“β€˜π‘€) β†’ (𝑧 βŠ† 𝑦 ↔ 𝑧 βŠ† (π‘“β€˜π‘€)))
5352rspcev 3607 . . . . . . . . . . . . . . . . . . 19 (((π‘“β€˜π‘€) ∈ ran 𝑓 ∧ 𝑧 βŠ† (π‘“β€˜π‘€)) β†’ βˆƒπ‘¦ ∈ ran 𝑓 𝑧 βŠ† 𝑦)
5451, 53sylan 579 . . . . . . . . . . . . . . . . . 18 (((𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) ∧ 𝑀 ∈ (cfβ€˜(β„΅β€˜π΄))) ∧ 𝑧 βŠ† (π‘“β€˜π‘€)) β†’ βˆƒπ‘¦ ∈ ran 𝑓 𝑧 βŠ† 𝑦)
5554rexlimdva2 3152 . . . . . . . . . . . . . . . . 17 (𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) β†’ (βˆƒπ‘€ ∈ (cfβ€˜(β„΅β€˜π΄))𝑧 βŠ† (π‘“β€˜π‘€) β†’ βˆƒπ‘¦ ∈ ran 𝑓 𝑧 βŠ† 𝑦))
5655ralimdv 3164 . . . . . . . . . . . . . . . 16 (𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) β†’ (βˆ€π‘§ ∈ (β„΅β€˜π΄)βˆƒπ‘€ ∈ (cfβ€˜(β„΅β€˜π΄))𝑧 βŠ† (π‘“β€˜π‘€) β†’ βˆ€π‘§ ∈ (β„΅β€˜π΄)βˆƒπ‘¦ ∈ ran 𝑓 𝑧 βŠ† 𝑦))
5756imp 406 . . . . . . . . . . . . . . 15 ((𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) ∧ βˆ€π‘§ ∈ (β„΅β€˜π΄)βˆƒπ‘€ ∈ (cfβ€˜(β„΅β€˜π΄))𝑧 βŠ† (π‘“β€˜π‘€)) β†’ βˆ€π‘§ ∈ (β„΅β€˜π΄)βˆƒπ‘¦ ∈ ran 𝑓 𝑧 βŠ† 𝑦)
5857adantl 481 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ (𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) ∧ βˆ€π‘§ ∈ (β„΅β€˜π΄)βˆƒπ‘€ ∈ (cfβ€˜(β„΅β€˜π΄))𝑧 βŠ† (π‘“β€˜π‘€))) β†’ βˆ€π‘§ ∈ (β„΅β€˜π΄)βˆƒπ‘¦ ∈ ran 𝑓 𝑧 βŠ† 𝑦)
59 alephislim 10098 . . . . . . . . . . . . . . . 16 (𝐴 ∈ On ↔ Lim (β„΅β€˜π΄))
6059biimpi 215 . . . . . . . . . . . . . . 15 (𝐴 ∈ On β†’ Lim (β„΅β€˜π΄))
61 frn 6723 . . . . . . . . . . . . . . . 16 (𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) β†’ ran 𝑓 βŠ† (β„΅β€˜π΄))
6261adantr 480 . . . . . . . . . . . . . . 15 ((𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) ∧ βˆ€π‘§ ∈ (β„΅β€˜π΄)βˆƒπ‘€ ∈ (cfβ€˜(β„΅β€˜π΄))𝑧 βŠ† (π‘“β€˜π‘€)) β†’ ran 𝑓 βŠ† (β„΅β€˜π΄))
63 coflim 10276 . . . . . . . . . . . . . . 15 ((Lim (β„΅β€˜π΄) ∧ ran 𝑓 βŠ† (β„΅β€˜π΄)) β†’ (βˆͺ ran 𝑓 = (β„΅β€˜π΄) ↔ βˆ€π‘§ ∈ (β„΅β€˜π΄)βˆƒπ‘¦ ∈ ran 𝑓 𝑧 βŠ† 𝑦))
6460, 62, 63syl2an 595 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ (𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) ∧ βˆ€π‘§ ∈ (β„΅β€˜π΄)βˆƒπ‘€ ∈ (cfβ€˜(β„΅β€˜π΄))𝑧 βŠ† (π‘“β€˜π‘€))) β†’ (βˆͺ ran 𝑓 = (β„΅β€˜π΄) ↔ βˆ€π‘§ ∈ (β„΅β€˜π΄)βˆƒπ‘¦ ∈ ran 𝑓 𝑧 βŠ† 𝑦))
6558, 64mpbird 257 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ (𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) ∧ βˆ€π‘§ ∈ (β„΅β€˜π΄)βˆƒπ‘€ ∈ (cfβ€˜(β„΅β€˜π΄))𝑧 βŠ† (π‘“β€˜π‘€))) β†’ βˆͺ ran 𝑓 = (β„΅β€˜π΄))
6649, 65eqtr3d 2769 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ (𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) ∧ βˆ€π‘§ ∈ (β„΅β€˜π΄)βˆƒπ‘€ ∈ (cfβ€˜(β„΅β€˜π΄))𝑧 βŠ† (π‘“β€˜π‘€))) β†’ βˆͺ π‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(π‘“β€˜π‘₯) = (β„΅β€˜π΄))
67 ffvelcdm 7085 . . . . . . . . . . . . . . . . 17 ((𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) ∧ π‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))) β†’ (π‘“β€˜π‘₯) ∈ (β„΅β€˜π΄))
6815oneli 6477 . . . . . . . . . . . . . . . . 17 ((π‘“β€˜π‘₯) ∈ (β„΅β€˜π΄) β†’ (π‘“β€˜π‘₯) ∈ On)
69 harcard 9993 . . . . . . . . . . . . . . . . . . 19 (cardβ€˜(harβ€˜(π‘“β€˜π‘₯))) = (harβ€˜(π‘“β€˜π‘₯))
70 iscard 9990 . . . . . . . . . . . . . . . . . . . 20 ((cardβ€˜(harβ€˜(π‘“β€˜π‘₯))) = (harβ€˜(π‘“β€˜π‘₯)) ↔ ((harβ€˜(π‘“β€˜π‘₯)) ∈ On ∧ βˆ€π‘¦ ∈ (harβ€˜(π‘“β€˜π‘₯))𝑦 β‰Ί (harβ€˜(π‘“β€˜π‘₯))))
7170simprbi 496 . . . . . . . . . . . . . . . . . . 19 ((cardβ€˜(harβ€˜(π‘“β€˜π‘₯))) = (harβ€˜(π‘“β€˜π‘₯)) β†’ βˆ€π‘¦ ∈ (harβ€˜(π‘“β€˜π‘₯))𝑦 β‰Ί (harβ€˜(π‘“β€˜π‘₯)))
7269, 71ax-mp 5 . . . . . . . . . . . . . . . . . 18 βˆ€π‘¦ ∈ (harβ€˜(π‘“β€˜π‘₯))𝑦 β‰Ί (harβ€˜(π‘“β€˜π‘₯))
73 domrefg 8999 . . . . . . . . . . . . . . . . . . . 20 ((π‘“β€˜π‘₯) ∈ V β†’ (π‘“β€˜π‘₯) β‰Ό (π‘“β€˜π‘₯))
7446, 73ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (π‘“β€˜π‘₯) β‰Ό (π‘“β€˜π‘₯)
75 elharval 9576 . . . . . . . . . . . . . . . . . . . 20 ((π‘“β€˜π‘₯) ∈ (harβ€˜(π‘“β€˜π‘₯)) ↔ ((π‘“β€˜π‘₯) ∈ On ∧ (π‘“β€˜π‘₯) β‰Ό (π‘“β€˜π‘₯)))
7675biimpri 227 . . . . . . . . . . . . . . . . . . 19 (((π‘“β€˜π‘₯) ∈ On ∧ (π‘“β€˜π‘₯) β‰Ό (π‘“β€˜π‘₯)) β†’ (π‘“β€˜π‘₯) ∈ (harβ€˜(π‘“β€˜π‘₯)))
7774, 76mpan2 690 . . . . . . . . . . . . . . . . . 18 ((π‘“β€˜π‘₯) ∈ On β†’ (π‘“β€˜π‘₯) ∈ (harβ€˜(π‘“β€˜π‘₯)))
78 breq1 5145 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (π‘“β€˜π‘₯) β†’ (𝑦 β‰Ί (harβ€˜(π‘“β€˜π‘₯)) ↔ (π‘“β€˜π‘₯) β‰Ί (harβ€˜(π‘“β€˜π‘₯))))
7978rspccv 3604 . . . . . . . . . . . . . . . . . 18 (βˆ€π‘¦ ∈ (harβ€˜(π‘“β€˜π‘₯))𝑦 β‰Ί (harβ€˜(π‘“β€˜π‘₯)) β†’ ((π‘“β€˜π‘₯) ∈ (harβ€˜(π‘“β€˜π‘₯)) β†’ (π‘“β€˜π‘₯) β‰Ί (harβ€˜(π‘“β€˜π‘₯))))
8072, 77, 79mpsyl 68 . . . . . . . . . . . . . . . . 17 ((π‘“β€˜π‘₯) ∈ On β†’ (π‘“β€˜π‘₯) β‰Ί (harβ€˜(π‘“β€˜π‘₯)))
8167, 68, 803syl 18 . . . . . . . . . . . . . . . 16 ((𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) ∧ π‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))) β†’ (π‘“β€˜π‘₯) β‰Ί (harβ€˜(π‘“β€˜π‘₯)))
82 harcl 9574 . . . . . . . . . . . . . . . . . . 19 (harβ€˜(π‘“β€˜π‘₯)) ∈ On
83 2fveq3 6896 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = π‘₯ β†’ (harβ€˜(π‘“β€˜π‘¦)) = (harβ€˜(π‘“β€˜π‘₯)))
84 pwcfsdom.1 . . . . . . . . . . . . . . . . . . . 20 𝐻 = (𝑦 ∈ (cfβ€˜(β„΅β€˜π΄)) ↦ (harβ€˜(π‘“β€˜π‘¦)))
8583, 84fvmptg 6997 . . . . . . . . . . . . . . . . . . 19 ((π‘₯ ∈ (cfβ€˜(β„΅β€˜π΄)) ∧ (harβ€˜(π‘“β€˜π‘₯)) ∈ On) β†’ (π»β€˜π‘₯) = (harβ€˜(π‘“β€˜π‘₯)))
8682, 85mpan2 690 . . . . . . . . . . . . . . . . . 18 (π‘₯ ∈ (cfβ€˜(β„΅β€˜π΄)) β†’ (π»β€˜π‘₯) = (harβ€˜(π‘“β€˜π‘₯)))
8786breq2d 5154 . . . . . . . . . . . . . . . . 17 (π‘₯ ∈ (cfβ€˜(β„΅β€˜π΄)) β†’ ((π‘“β€˜π‘₯) β‰Ί (π»β€˜π‘₯) ↔ (π‘“β€˜π‘₯) β‰Ί (harβ€˜(π‘“β€˜π‘₯))))
8887adantl 481 . . . . . . . . . . . . . . . 16 ((𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) ∧ π‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))) β†’ ((π‘“β€˜π‘₯) β‰Ί (π»β€˜π‘₯) ↔ (π‘“β€˜π‘₯) β‰Ί (harβ€˜(π‘“β€˜π‘₯))))
8981, 88mpbird 257 . . . . . . . . . . . . . . 15 ((𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) ∧ π‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))) β†’ (π‘“β€˜π‘₯) β‰Ί (π»β€˜π‘₯))
9089ralrimiva 3141 . . . . . . . . . . . . . 14 (𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) β†’ βˆ€π‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(π‘“β€˜π‘₯) β‰Ί (π»β€˜π‘₯))
91 fvex 6904 . . . . . . . . . . . . . . 15 (cfβ€˜(β„΅β€˜π΄)) ∈ V
92 eqid 2727 . . . . . . . . . . . . . . 15 βˆͺ π‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(π‘“β€˜π‘₯) = βˆͺ π‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(π‘“β€˜π‘₯)
93 eqid 2727 . . . . . . . . . . . . . . 15 Xπ‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(π»β€˜π‘₯) = Xπ‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(π»β€˜π‘₯)
9491, 92, 93konigth 10584 . . . . . . . . . . . . . 14 (βˆ€π‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(π‘“β€˜π‘₯) β‰Ί (π»β€˜π‘₯) β†’ βˆͺ π‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(π‘“β€˜π‘₯) β‰Ί Xπ‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(π»β€˜π‘₯))
9590, 94syl 17 . . . . . . . . . . . . 13 (𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) β†’ βˆͺ π‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(π‘“β€˜π‘₯) β‰Ί Xπ‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(π»β€˜π‘₯))
9695ad2antrl 727 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ (𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) ∧ βˆ€π‘§ ∈ (β„΅β€˜π΄)βˆƒπ‘€ ∈ (cfβ€˜(β„΅β€˜π΄))𝑧 βŠ† (π‘“β€˜π‘€))) β†’ βˆͺ π‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(π‘“β€˜π‘₯) β‰Ί Xπ‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(π»β€˜π‘₯))
9766, 96eqbrtrrd 5166 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ (𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) ∧ βˆ€π‘§ ∈ (β„΅β€˜π΄)βˆƒπ‘€ ∈ (cfβ€˜(β„΅β€˜π΄))𝑧 βŠ† (π‘“β€˜π‘€))) β†’ (β„΅β€˜π΄) β‰Ί Xπ‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(π»β€˜π‘₯))
9841, 97sylan 579 . . . . . . . . . 10 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ (𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) ∧ βˆ€π‘§ ∈ (β„΅β€˜π΄)βˆƒπ‘€ ∈ (cfβ€˜(β„΅β€˜π΄))𝑧 βŠ† (π‘“β€˜π‘€))) β†’ (β„΅β€˜π΄) β‰Ί Xπ‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(π»β€˜π‘₯))
99 ovex 7447 . . . . . . . . . . . 12 ((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄))) ∈ V
10067ex 412 . . . . . . . . . . . . . . . 16 (𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) β†’ (π‘₯ ∈ (cfβ€˜(β„΅β€˜π΄)) β†’ (π‘“β€˜π‘₯) ∈ (β„΅β€˜π΄)))
101 alephlim 10082 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ V ∧ Lim 𝐴) β†’ (β„΅β€˜π΄) = βˆͺ 𝑦 ∈ 𝐴 (β„΅β€˜π‘¦))
102101eleq2d 2814 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ V ∧ Lim 𝐴) β†’ ((π‘“β€˜π‘₯) ∈ (β„΅β€˜π΄) ↔ (π‘“β€˜π‘₯) ∈ βˆͺ 𝑦 ∈ 𝐴 (β„΅β€˜π‘¦)))
103 eliun 4995 . . . . . . . . . . . . . . . . . . 19 ((π‘“β€˜π‘₯) ∈ βˆͺ 𝑦 ∈ 𝐴 (β„΅β€˜π‘¦) ↔ βˆƒπ‘¦ ∈ 𝐴 (π‘“β€˜π‘₯) ∈ (β„΅β€˜π‘¦))
104 alephcard 10085 . . . . . . . . . . . . . . . . . . . . . . . 24 (cardβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦)
105104eleq2i 2820 . . . . . . . . . . . . . . . . . . . . . . 23 ((π‘“β€˜π‘₯) ∈ (cardβ€˜(β„΅β€˜π‘¦)) ↔ (π‘“β€˜π‘₯) ∈ (β„΅β€˜π‘¦))
106 cardsdomelir 9988 . . . . . . . . . . . . . . . . . . . . . . 23 ((π‘“β€˜π‘₯) ∈ (cardβ€˜(β„΅β€˜π‘¦)) β†’ (π‘“β€˜π‘₯) β‰Ί (β„΅β€˜π‘¦))
107105, 106sylbir 234 . . . . . . . . . . . . . . . . . . . . . 22 ((π‘“β€˜π‘₯) ∈ (β„΅β€˜π‘¦) β†’ (π‘“β€˜π‘₯) β‰Ί (β„΅β€˜π‘¦))
108 elharval 9576 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((β„΅β€˜π‘¦) ∈ (harβ€˜(π‘“β€˜π‘₯)) ↔ ((β„΅β€˜π‘¦) ∈ On ∧ (β„΅β€˜π‘¦) β‰Ό (π‘“β€˜π‘₯)))
109108simprbi 496 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((β„΅β€˜π‘¦) ∈ (harβ€˜(π‘“β€˜π‘₯)) β†’ (β„΅β€˜π‘¦) β‰Ό (π‘“β€˜π‘₯))
110 domnsym 9115 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((β„΅β€˜π‘¦) β‰Ό (π‘“β€˜π‘₯) β†’ Β¬ (π‘“β€˜π‘₯) β‰Ί (β„΅β€˜π‘¦))
111109, 110syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((β„΅β€˜π‘¦) ∈ (harβ€˜(π‘“β€˜π‘₯)) β†’ Β¬ (π‘“β€˜π‘₯) β‰Ί (β„΅β€˜π‘¦))
112111con2i 139 . . . . . . . . . . . . . . . . . . . . . . 23 ((π‘“β€˜π‘₯) β‰Ί (β„΅β€˜π‘¦) β†’ Β¬ (β„΅β€˜π‘¦) ∈ (harβ€˜(π‘“β€˜π‘₯)))
113 alephon 10084 . . . . . . . . . . . . . . . . . . . . . . . 24 (β„΅β€˜π‘¦) ∈ On
114 ontri1 6397 . . . . . . . . . . . . . . . . . . . . . . . 24 (((harβ€˜(π‘“β€˜π‘₯)) ∈ On ∧ (β„΅β€˜π‘¦) ∈ On) β†’ ((harβ€˜(π‘“β€˜π‘₯)) βŠ† (β„΅β€˜π‘¦) ↔ Β¬ (β„΅β€˜π‘¦) ∈ (harβ€˜(π‘“β€˜π‘₯))))
11582, 113, 114mp2an 691 . . . . . . . . . . . . . . . . . . . . . . 23 ((harβ€˜(π‘“β€˜π‘₯)) βŠ† (β„΅β€˜π‘¦) ↔ Β¬ (β„΅β€˜π‘¦) ∈ (harβ€˜(π‘“β€˜π‘₯)))
116112, 115sylibr 233 . . . . . . . . . . . . . . . . . . . . . 22 ((π‘“β€˜π‘₯) β‰Ί (β„΅β€˜π‘¦) β†’ (harβ€˜(π‘“β€˜π‘₯)) βŠ† (β„΅β€˜π‘¦))
117107, 116syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((π‘“β€˜π‘₯) ∈ (β„΅β€˜π‘¦) β†’ (harβ€˜(π‘“β€˜π‘₯)) βŠ† (β„΅β€˜π‘¦))
118 alephord2i 10092 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴 ∈ On β†’ (𝑦 ∈ 𝐴 β†’ (β„΅β€˜π‘¦) ∈ (β„΅β€˜π΄)))
119118imp 406 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) β†’ (β„΅β€˜π‘¦) ∈ (β„΅β€˜π΄))
120 ontr2 6410 . . . . . . . . . . . . . . . . . . . . . 22 (((harβ€˜(π‘“β€˜π‘₯)) ∈ On ∧ (β„΅β€˜π΄) ∈ On) β†’ (((harβ€˜(π‘“β€˜π‘₯)) βŠ† (β„΅β€˜π‘¦) ∧ (β„΅β€˜π‘¦) ∈ (β„΅β€˜π΄)) β†’ (harβ€˜(π‘“β€˜π‘₯)) ∈ (β„΅β€˜π΄)))
12182, 15, 120mp2an 691 . . . . . . . . . . . . . . . . . . . . 21 (((harβ€˜(π‘“β€˜π‘₯)) βŠ† (β„΅β€˜π‘¦) ∧ (β„΅β€˜π‘¦) ∈ (β„΅β€˜π΄)) β†’ (harβ€˜(π‘“β€˜π‘₯)) ∈ (β„΅β€˜π΄))
122117, 119, 121syl2anr 596 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ (π‘“β€˜π‘₯) ∈ (β„΅β€˜π‘¦)) β†’ (harβ€˜(π‘“β€˜π‘₯)) ∈ (β„΅β€˜π΄))
123122rexlimdva2 3152 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ On β†’ (βˆƒπ‘¦ ∈ 𝐴 (π‘“β€˜π‘₯) ∈ (β„΅β€˜π‘¦) β†’ (harβ€˜(π‘“β€˜π‘₯)) ∈ (β„΅β€˜π΄)))
124103, 123biimtrid 241 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ On β†’ ((π‘“β€˜π‘₯) ∈ βˆͺ 𝑦 ∈ 𝐴 (β„΅β€˜π‘¦) β†’ (harβ€˜(π‘“β€˜π‘₯)) ∈ (β„΅β€˜π΄)))
12541, 124syl 17 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ V ∧ Lim 𝐴) β†’ ((π‘“β€˜π‘₯) ∈ βˆͺ 𝑦 ∈ 𝐴 (β„΅β€˜π‘¦) β†’ (harβ€˜(π‘“β€˜π‘₯)) ∈ (β„΅β€˜π΄)))
126102, 125sylbid 239 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ V ∧ Lim 𝐴) β†’ ((π‘“β€˜π‘₯) ∈ (β„΅β€˜π΄) β†’ (harβ€˜(π‘“β€˜π‘₯)) ∈ (β„΅β€˜π΄)))
127100, 126sylan9r 508 . . . . . . . . . . . . . . 15 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄)) β†’ (π‘₯ ∈ (cfβ€˜(β„΅β€˜π΄)) β†’ (harβ€˜(π‘“β€˜π‘₯)) ∈ (β„΅β€˜π΄)))
128127imp 406 . . . . . . . . . . . . . 14 ((((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄)) ∧ π‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))) β†’ (harβ€˜(π‘“β€˜π‘₯)) ∈ (β„΅β€˜π΄))
12983cbvmptv 5255 . . . . . . . . . . . . . . 15 (𝑦 ∈ (cfβ€˜(β„΅β€˜π΄)) ↦ (harβ€˜(π‘“β€˜π‘¦))) = (π‘₯ ∈ (cfβ€˜(β„΅β€˜π΄)) ↦ (harβ€˜(π‘“β€˜π‘₯)))
13084, 129eqtri 2755 . . . . . . . . . . . . . 14 𝐻 = (π‘₯ ∈ (cfβ€˜(β„΅β€˜π΄)) ↦ (harβ€˜(π‘“β€˜π‘₯)))
131128, 130fmptd 7118 . . . . . . . . . . . . 13 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄)) β†’ 𝐻:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄))
132 ffvelcdm 7085 . . . . . . . . . . . . . . 15 ((𝐻:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) ∧ π‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))) β†’ (π»β€˜π‘₯) ∈ (β„΅β€˜π΄))
133 onelss 6405 . . . . . . . . . . . . . . 15 ((β„΅β€˜π΄) ∈ On β†’ ((π»β€˜π‘₯) ∈ (β„΅β€˜π΄) β†’ (π»β€˜π‘₯) βŠ† (β„΅β€˜π΄)))
13415, 132, 133mpsyl 68 . . . . . . . . . . . . . 14 ((𝐻:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) ∧ π‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))) β†’ (π»β€˜π‘₯) βŠ† (β„΅β€˜π΄))
135134ralrimiva 3141 . . . . . . . . . . . . 13 (𝐻:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) β†’ βˆ€π‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(π»β€˜π‘₯) βŠ† (β„΅β€˜π΄))
136 ss2ixp 8920 . . . . . . . . . . . . . 14 (βˆ€π‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(π»β€˜π‘₯) βŠ† (β„΅β€˜π΄) β†’ Xπ‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(π»β€˜π‘₯) βŠ† Xπ‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(β„΅β€˜π΄))
13791, 10ixpconst 8917 . . . . . . . . . . . . . 14 Xπ‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(β„΅β€˜π΄) = ((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄)))
138136, 137sseqtrdi 4028 . . . . . . . . . . . . 13 (βˆ€π‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(π»β€˜π‘₯) βŠ† (β„΅β€˜π΄) β†’ Xπ‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(π»β€˜π‘₯) βŠ† ((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄))))
139131, 135, 1383syl 18 . . . . . . . . . . . 12 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄)) β†’ Xπ‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(π»β€˜π‘₯) βŠ† ((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄))))
140 ssdomg 9012 . . . . . . . . . . . 12 (((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄))) ∈ V β†’ (Xπ‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(π»β€˜π‘₯) βŠ† ((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄))) β†’ Xπ‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(π»β€˜π‘₯) β‰Ό ((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄)))))
14199, 139, 140mpsyl 68 . . . . . . . . . . 11 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄)) β†’ Xπ‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(π»β€˜π‘₯) β‰Ό ((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄))))
142141adantrr 716 . . . . . . . . . 10 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ (𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) ∧ βˆ€π‘§ ∈ (β„΅β€˜π΄)βˆƒπ‘€ ∈ (cfβ€˜(β„΅β€˜π΄))𝑧 βŠ† (π‘“β€˜π‘€))) β†’ Xπ‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(π»β€˜π‘₯) β‰Ό ((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄))))
143 sdomdomtr 9126 . . . . . . . . . 10 (((β„΅β€˜π΄) β‰Ί Xπ‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(π»β€˜π‘₯) ∧ Xπ‘₯ ∈ (cfβ€˜(β„΅β€˜π΄))(π»β€˜π‘₯) β‰Ό ((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄)))) β†’ (β„΅β€˜π΄) β‰Ί ((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄))))
14498, 142, 143syl2anc 583 . . . . . . . . 9 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ (𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) ∧ βˆ€π‘§ ∈ (β„΅β€˜π΄)βˆƒπ‘€ ∈ (cfβ€˜(β„΅β€˜π΄))𝑧 βŠ† (π‘“β€˜π‘€))) β†’ (β„΅β€˜π΄) β‰Ί ((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄))))
145144expcom 413 . . . . . . . 8 ((𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) ∧ βˆ€π‘§ ∈ (β„΅β€˜π΄)βˆƒπ‘€ ∈ (cfβ€˜(β„΅β€˜π΄))𝑧 βŠ† (π‘“β€˜π‘€)) β†’ ((𝐴 ∈ V ∧ Lim 𝐴) β†’ (β„΅β€˜π΄) β‰Ί ((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄)))))
1461453adant2 1129 . . . . . . 7 ((𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) ∧ Smo 𝑓 ∧ βˆ€π‘§ ∈ (β„΅β€˜π΄)βˆƒπ‘€ ∈ (cfβ€˜(β„΅β€˜π΄))𝑧 βŠ† (π‘“β€˜π‘€)) β†’ ((𝐴 ∈ V ∧ Lim 𝐴) β†’ (β„΅β€˜π΄) β‰Ί ((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄)))))
147146exlimiv 1926 . . . . . 6 (βˆƒπ‘“(𝑓:(cfβ€˜(β„΅β€˜π΄))⟢(β„΅β€˜π΄) ∧ Smo 𝑓 ∧ βˆ€π‘§ ∈ (β„΅β€˜π΄)βˆƒπ‘€ ∈ (cfβ€˜(β„΅β€˜π΄))𝑧 βŠ† (π‘“β€˜π‘€)) β†’ ((𝐴 ∈ V ∧ Lim 𝐴) β†’ (β„΅β€˜π΄) β‰Ί ((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄)))))
14815, 40, 147mp2b 10 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) β†’ (β„΅β€˜π΄) β‰Ί ((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄))))
149148a1i 11 . . . 4 (𝐴 ∈ On β†’ ((𝐴 ∈ V ∧ Lim 𝐴) β†’ (β„΅β€˜π΄) β‰Ί ((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄)))))
15033, 39, 1493jaod 1426 . . 3 (𝐴 ∈ On β†’ ((𝐴 = βˆ… ∨ βˆƒπ‘₯ ∈ On 𝐴 = suc π‘₯ ∨ (𝐴 ∈ V ∧ Lim 𝐴)) β†’ (β„΅β€˜π΄) β‰Ί ((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄)))))
1512, 150mpd 15 . 2 (𝐴 ∈ On β†’ (β„΅β€˜π΄) β‰Ί ((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄))))
152 alephfnon 10080 . . . . 5 β„΅ Fn On
153152fndmi 6652 . . . 4 dom β„΅ = On
154153eleq2i 2820 . . 3 (𝐴 ∈ dom β„΅ ↔ 𝐴 ∈ On)
155 ndmfv 6926 . . . 4 (Β¬ 𝐴 ∈ dom β„΅ β†’ (β„΅β€˜π΄) = βˆ…)
156 1n0 8502 . . . . . 6 1o β‰  βˆ…
157 1oex 8490 . . . . . . 7 1o ∈ V
1581570sdom 9123 . . . . . 6 (βˆ… β‰Ί 1o ↔ 1o β‰  βˆ…)
159156, 158mpbir 230 . . . . 5 βˆ… β‰Ί 1o
160 id 22 . . . . . 6 ((β„΅β€˜π΄) = βˆ… β†’ (β„΅β€˜π΄) = βˆ…)
161 fveq2 6891 . . . . . . . . 9 ((β„΅β€˜π΄) = βˆ… β†’ (cfβ€˜(β„΅β€˜π΄)) = (cfβ€˜βˆ…))
162 cf0 10266 . . . . . . . . 9 (cfβ€˜βˆ…) = βˆ…
163161, 162eqtrdi 2783 . . . . . . . 8 ((β„΅β€˜π΄) = βˆ… β†’ (cfβ€˜(β„΅β€˜π΄)) = βˆ…)
164160, 163oveq12d 7432 . . . . . . 7 ((β„΅β€˜π΄) = βˆ… β†’ ((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄))) = (βˆ… ↑m βˆ…))
165 0ex 5301 . . . . . . . 8 βˆ… ∈ V
166 map0e 8892 . . . . . . . 8 (βˆ… ∈ V β†’ (βˆ… ↑m βˆ…) = 1o)
167165, 166ax-mp 5 . . . . . . 7 (βˆ… ↑m βˆ…) = 1o
168164, 167eqtrdi 2783 . . . . . 6 ((β„΅β€˜π΄) = βˆ… β†’ ((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄))) = 1o)
169160, 168breq12d 5155 . . . . 5 ((β„΅β€˜π΄) = βˆ… β†’ ((β„΅β€˜π΄) β‰Ί ((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄))) ↔ βˆ… β‰Ί 1o))
170159, 169mpbiri 258 . . . 4 ((β„΅β€˜π΄) = βˆ… β†’ (β„΅β€˜π΄) β‰Ί ((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄))))
171155, 170syl 17 . . 3 (Β¬ 𝐴 ∈ dom β„΅ β†’ (β„΅β€˜π΄) β‰Ί ((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄))))
172154, 171sylnbir 331 . 2 (Β¬ 𝐴 ∈ On β†’ (β„΅β€˜π΄) β‰Ί ((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄))))
173151, 172pm2.61i 182 1 (β„΅β€˜π΄) β‰Ί ((β„΅β€˜π΄) ↑m (cfβ€˜(β„΅β€˜π΄)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∨ w3o 1084   ∧ w3a 1085   = wceq 1534  βˆƒwex 1774   ∈ wcel 2099  {cab 2704   β‰  wne 2935  βˆ€wral 3056  βˆƒwrex 3065  Vcvv 3469   βŠ† wss 3944  βˆ…c0 4318  π’« cpw 4598  βˆͺ cuni 4903  βˆͺ ciun 4991   class class class wbr 5142   ↦ cmpt 5225  dom cdm 5672  ran crn 5673  Oncon0 6363  Lim wlim 6364  suc csuc 6365   Fn wfn 6537  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7414  Ο‰com 7864  Smo wsmo 8359  1oc1o 8473  2oc2o 8474   ↑m cmap 8836  Xcixp 8907   β‰ˆ cen 8952   β‰Ό cdom 8953   β‰Ί csdm 8954  harchar 9571  cardccrd 9950  β„΅cale 9951  cfccf 9952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9656  ax-ac2 10478
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-smo 8360  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8718  df-map 8838  df-ixp 8908  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-oi 9525  df-har 9572  df-card 9954  df-aleph 9955  df-cf 9956  df-acn 9957  df-ac 10131
This theorem is referenced by:  cfpwsdom  10599  tskcard  10796  bj-pwcfsdom  36477
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