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Theorem infpwfien 10005
Description: Any infinite well-orderable set is equinumerous to its set of finite subsets. (Contributed by Mario Carneiro, 18-May-2015.)
Assertion
Ref Expression
infpwfien ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ (𝒫 𝐴 ∩ Fin) β‰ˆ 𝐴)

Proof of Theorem infpwfien
Dummy variables π‘š 𝑛 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 infxpidm2 9960 . . . . . . 7 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ (𝐴 Γ— 𝐴) β‰ˆ 𝐴)
2 infn0 9258 . . . . . . . 8 (Ο‰ β‰Ό 𝐴 β†’ 𝐴 β‰  βˆ…)
32adantl 483 . . . . . . 7 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ 𝐴 β‰  βˆ…)
4 fseqen 9970 . . . . . . 7 (((𝐴 Γ— 𝐴) β‰ˆ 𝐴 ∧ 𝐴 β‰  βˆ…) β†’ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) β‰ˆ (Ο‰ Γ— 𝐴))
51, 3, 4syl2anc 585 . . . . . 6 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) β‰ˆ (Ο‰ Γ— 𝐴))
6 xpdom1g 9020 . . . . . . 7 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ (Ο‰ Γ— 𝐴) β‰Ό (𝐴 Γ— 𝐴))
7 domentr 8960 . . . . . . 7 (((Ο‰ Γ— 𝐴) β‰Ό (𝐴 Γ— 𝐴) ∧ (𝐴 Γ— 𝐴) β‰ˆ 𝐴) β†’ (Ο‰ Γ— 𝐴) β‰Ό 𝐴)
86, 1, 7syl2anc 585 . . . . . 6 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ (Ο‰ Γ— 𝐴) β‰Ό 𝐴)
9 endomtr 8959 . . . . . 6 ((βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) β‰ˆ (Ο‰ Γ— 𝐴) ∧ (Ο‰ Γ— 𝐴) β‰Ό 𝐴) β†’ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) β‰Ό 𝐴)
105, 8, 9syl2anc 585 . . . . 5 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) β‰Ό 𝐴)
11 numdom 9981 . . . . 5 ((𝐴 ∈ dom card ∧ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) β‰Ό 𝐴) β†’ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ∈ dom card)
1210, 11syldan 592 . . . 4 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ∈ dom card)
13 eliun 4963 . . . . . . . . 9 (π‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ↔ βˆƒπ‘› ∈ Ο‰ π‘₯ ∈ (𝐴 ↑m 𝑛))
14 elmapi 8794 . . . . . . . . . . . . . . 15 (π‘₯ ∈ (𝐴 ↑m 𝑛) β†’ π‘₯:π‘›βŸΆπ΄)
1514ad2antll 728 . . . . . . . . . . . . . 14 (((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ (𝑛 ∈ Ο‰ ∧ π‘₯ ∈ (𝐴 ↑m 𝑛))) β†’ π‘₯:π‘›βŸΆπ΄)
1615frnd 6681 . . . . . . . . . . . . 13 (((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ (𝑛 ∈ Ο‰ ∧ π‘₯ ∈ (𝐴 ↑m 𝑛))) β†’ ran π‘₯ βŠ† 𝐴)
17 vex 3452 . . . . . . . . . . . . . . 15 π‘₯ ∈ V
1817rnex 7854 . . . . . . . . . . . . . 14 ran π‘₯ ∈ V
1918elpw 4569 . . . . . . . . . . . . 13 (ran π‘₯ ∈ 𝒫 𝐴 ↔ ran π‘₯ βŠ† 𝐴)
2016, 19sylibr 233 . . . . . . . . . . . 12 (((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ (𝑛 ∈ Ο‰ ∧ π‘₯ ∈ (𝐴 ↑m 𝑛))) β†’ ran π‘₯ ∈ 𝒫 𝐴)
21 simprl 770 . . . . . . . . . . . . . 14 (((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ (𝑛 ∈ Ο‰ ∧ π‘₯ ∈ (𝐴 ↑m 𝑛))) β†’ 𝑛 ∈ Ο‰)
22 ssid 3971 . . . . . . . . . . . . . 14 𝑛 βŠ† 𝑛
23 ssnnfi 9120 . . . . . . . . . . . . . 14 ((𝑛 ∈ Ο‰ ∧ 𝑛 βŠ† 𝑛) β†’ 𝑛 ∈ Fin)
2421, 22, 23sylancl 587 . . . . . . . . . . . . 13 (((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ (𝑛 ∈ Ο‰ ∧ π‘₯ ∈ (𝐴 ↑m 𝑛))) β†’ 𝑛 ∈ Fin)
25 ffn 6673 . . . . . . . . . . . . . . 15 (π‘₯:π‘›βŸΆπ΄ β†’ π‘₯ Fn 𝑛)
26 dffn4 6767 . . . . . . . . . . . . . . 15 (π‘₯ Fn 𝑛 ↔ π‘₯:𝑛–ontoβ†’ran π‘₯)
2725, 26sylib 217 . . . . . . . . . . . . . 14 (π‘₯:π‘›βŸΆπ΄ β†’ π‘₯:𝑛–ontoβ†’ran π‘₯)
2815, 27syl 17 . . . . . . . . . . . . 13 (((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ (𝑛 ∈ Ο‰ ∧ π‘₯ ∈ (𝐴 ↑m 𝑛))) β†’ π‘₯:𝑛–ontoβ†’ran π‘₯)
29 fofi 9289 . . . . . . . . . . . . 13 ((𝑛 ∈ Fin ∧ π‘₯:𝑛–ontoβ†’ran π‘₯) β†’ ran π‘₯ ∈ Fin)
3024, 28, 29syl2anc 585 . . . . . . . . . . . 12 (((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ (𝑛 ∈ Ο‰ ∧ π‘₯ ∈ (𝐴 ↑m 𝑛))) β†’ ran π‘₯ ∈ Fin)
3120, 30elind 4159 . . . . . . . . . . 11 (((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ (𝑛 ∈ Ο‰ ∧ π‘₯ ∈ (𝐴 ↑m 𝑛))) β†’ ran π‘₯ ∈ (𝒫 𝐴 ∩ Fin))
3231expr 458 . . . . . . . . . 10 (((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ 𝑛 ∈ Ο‰) β†’ (π‘₯ ∈ (𝐴 ↑m 𝑛) β†’ ran π‘₯ ∈ (𝒫 𝐴 ∩ Fin)))
3332rexlimdva 3153 . . . . . . . . 9 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ (βˆƒπ‘› ∈ Ο‰ π‘₯ ∈ (𝐴 ↑m 𝑛) β†’ ran π‘₯ ∈ (𝒫 𝐴 ∩ Fin)))
3413, 33biimtrid 241 . . . . . . . 8 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ (π‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) β†’ ran π‘₯ ∈ (𝒫 𝐴 ∩ Fin)))
3534imp 408 . . . . . . 7 (((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ π‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛)) β†’ ran π‘₯ ∈ (𝒫 𝐴 ∩ Fin))
3635fmpttd 7068 . . . . . 6 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ (π‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ↦ ran π‘₯):βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛)⟢(𝒫 𝐴 ∩ Fin))
3736ffnd 6674 . . . . 5 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ (π‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ↦ ran π‘₯) Fn βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛))
3836frnd 6681 . . . . . 6 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ ran (π‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ↦ ran π‘₯) βŠ† (𝒫 𝐴 ∩ Fin))
39 simpr 486 . . . . . . . . . . . 12 (((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) β†’ 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
4039elin2d 4164 . . . . . . . . . . 11 (((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) β†’ 𝑦 ∈ Fin)
41 isfi 8923 . . . . . . . . . . 11 (𝑦 ∈ Fin ↔ βˆƒπ‘š ∈ Ο‰ 𝑦 β‰ˆ π‘š)
4240, 41sylib 217 . . . . . . . . . 10 (((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) β†’ βˆƒπ‘š ∈ Ο‰ 𝑦 β‰ˆ π‘š)
43 ensym 8950 . . . . . . . . . . . . 13 (𝑦 β‰ˆ π‘š β†’ π‘š β‰ˆ 𝑦)
44 bren 8900 . . . . . . . . . . . . 13 (π‘š β‰ˆ 𝑦 ↔ βˆƒπ‘₯ π‘₯:π‘šβ€“1-1-onto→𝑦)
4543, 44sylib 217 . . . . . . . . . . . 12 (𝑦 β‰ˆ π‘š β†’ βˆƒπ‘₯ π‘₯:π‘šβ€“1-1-onto→𝑦)
46 simprl 770 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (π‘š ∈ Ο‰ ∧ π‘₯:π‘šβ€“1-1-onto→𝑦)) β†’ π‘š ∈ Ο‰)
47 f1of 6789 . . . . . . . . . . . . . . . . . . . 20 (π‘₯:π‘šβ€“1-1-onto→𝑦 β†’ π‘₯:π‘šβŸΆπ‘¦)
4847ad2antll 728 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (π‘š ∈ Ο‰ ∧ π‘₯:π‘šβ€“1-1-onto→𝑦)) β†’ π‘₯:π‘šβŸΆπ‘¦)
49 simplr 768 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (π‘š ∈ Ο‰ ∧ π‘₯:π‘šβ€“1-1-onto→𝑦)) β†’ 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
5049elin1d 4163 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (π‘š ∈ Ο‰ ∧ π‘₯:π‘šβ€“1-1-onto→𝑦)) β†’ 𝑦 ∈ 𝒫 𝐴)
5150elpwid 4574 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (π‘š ∈ Ο‰ ∧ π‘₯:π‘šβ€“1-1-onto→𝑦)) β†’ 𝑦 βŠ† 𝐴)
5248, 51fssd 6691 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (π‘š ∈ Ο‰ ∧ π‘₯:π‘šβ€“1-1-onto→𝑦)) β†’ π‘₯:π‘šβŸΆπ΄)
53 simplll 774 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (π‘š ∈ Ο‰ ∧ π‘₯:π‘šβ€“1-1-onto→𝑦)) β†’ 𝐴 ∈ dom card)
54 vex 3452 . . . . . . . . . . . . . . . . . . 19 π‘š ∈ V
55 elmapg 8785 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ dom card ∧ π‘š ∈ V) β†’ (π‘₯ ∈ (𝐴 ↑m π‘š) ↔ π‘₯:π‘šβŸΆπ΄))
5653, 54, 55sylancl 587 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (π‘š ∈ Ο‰ ∧ π‘₯:π‘šβ€“1-1-onto→𝑦)) β†’ (π‘₯ ∈ (𝐴 ↑m π‘š) ↔ π‘₯:π‘šβŸΆπ΄))
5752, 56mpbird 257 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (π‘š ∈ Ο‰ ∧ π‘₯:π‘šβ€“1-1-onto→𝑦)) β†’ π‘₯ ∈ (𝐴 ↑m π‘š))
58 oveq2 7370 . . . . . . . . . . . . . . . . . . 19 (𝑛 = π‘š β†’ (𝐴 ↑m 𝑛) = (𝐴 ↑m π‘š))
5958eleq2d 2824 . . . . . . . . . . . . . . . . . 18 (𝑛 = π‘š β†’ (π‘₯ ∈ (𝐴 ↑m 𝑛) ↔ π‘₯ ∈ (𝐴 ↑m π‘š)))
6059rspcev 3584 . . . . . . . . . . . . . . . . 17 ((π‘š ∈ Ο‰ ∧ π‘₯ ∈ (𝐴 ↑m π‘š)) β†’ βˆƒπ‘› ∈ Ο‰ π‘₯ ∈ (𝐴 ↑m 𝑛))
6146, 57, 60syl2anc 585 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (π‘š ∈ Ο‰ ∧ π‘₯:π‘šβ€“1-1-onto→𝑦)) β†’ βˆƒπ‘› ∈ Ο‰ π‘₯ ∈ (𝐴 ↑m 𝑛))
6261, 13sylibr 233 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (π‘š ∈ Ο‰ ∧ π‘₯:π‘šβ€“1-1-onto→𝑦)) β†’ π‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛))
63 f1ofo 6796 . . . . . . . . . . . . . . . . . 18 (π‘₯:π‘šβ€“1-1-onto→𝑦 β†’ π‘₯:π‘šβ€“onto→𝑦)
6463ad2antll 728 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (π‘š ∈ Ο‰ ∧ π‘₯:π‘šβ€“1-1-onto→𝑦)) β†’ π‘₯:π‘šβ€“onto→𝑦)
65 forn 6764 . . . . . . . . . . . . . . . . 17 (π‘₯:π‘šβ€“onto→𝑦 β†’ ran π‘₯ = 𝑦)
6664, 65syl 17 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (π‘š ∈ Ο‰ ∧ π‘₯:π‘šβ€“1-1-onto→𝑦)) β†’ ran π‘₯ = 𝑦)
6766eqcomd 2743 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (π‘š ∈ Ο‰ ∧ π‘₯:π‘šβ€“1-1-onto→𝑦)) β†’ 𝑦 = ran π‘₯)
6862, 67jca 513 . . . . . . . . . . . . . 14 ((((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (π‘š ∈ Ο‰ ∧ π‘₯:π‘šβ€“1-1-onto→𝑦)) β†’ (π‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ∧ 𝑦 = ran π‘₯))
6968expr 458 . . . . . . . . . . . . 13 ((((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ π‘š ∈ Ο‰) β†’ (π‘₯:π‘šβ€“1-1-onto→𝑦 β†’ (π‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ∧ 𝑦 = ran π‘₯)))
7069eximdv 1921 . . . . . . . . . . . 12 ((((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ π‘š ∈ Ο‰) β†’ (βˆƒπ‘₯ π‘₯:π‘šβ€“1-1-onto→𝑦 β†’ βˆƒπ‘₯(π‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ∧ 𝑦 = ran π‘₯)))
7145, 70syl5 34 . . . . . . . . . . 11 ((((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ π‘š ∈ Ο‰) β†’ (𝑦 β‰ˆ π‘š β†’ βˆƒπ‘₯(π‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ∧ 𝑦 = ran π‘₯)))
7271rexlimdva 3153 . . . . . . . . . 10 (((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) β†’ (βˆƒπ‘š ∈ Ο‰ 𝑦 β‰ˆ π‘š β†’ βˆƒπ‘₯(π‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ∧ 𝑦 = ran π‘₯)))
7342, 72mpd 15 . . . . . . . . 9 (((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) β†’ βˆƒπ‘₯(π‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ∧ 𝑦 = ran π‘₯))
7473ex 414 . . . . . . . 8 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ (𝑦 ∈ (𝒫 𝐴 ∩ Fin) β†’ βˆƒπ‘₯(π‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ∧ 𝑦 = ran π‘₯)))
75 eqid 2737 . . . . . . . . . . 11 (π‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ↦ ran π‘₯) = (π‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ↦ ran π‘₯)
7675elrnmpt 5916 . . . . . . . . . 10 (𝑦 ∈ V β†’ (𝑦 ∈ ran (π‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ↦ ran π‘₯) ↔ βˆƒπ‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛)𝑦 = ran π‘₯))
7776elv 3454 . . . . . . . . 9 (𝑦 ∈ ran (π‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ↦ ran π‘₯) ↔ βˆƒπ‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛)𝑦 = ran π‘₯)
78 df-rex 3075 . . . . . . . . 9 (βˆƒπ‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛)𝑦 = ran π‘₯ ↔ βˆƒπ‘₯(π‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ∧ 𝑦 = ran π‘₯))
7977, 78bitri 275 . . . . . . . 8 (𝑦 ∈ ran (π‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ↦ ran π‘₯) ↔ βˆƒπ‘₯(π‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ∧ 𝑦 = ran π‘₯))
8074, 79syl6ibr 252 . . . . . . 7 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ (𝑦 ∈ (𝒫 𝐴 ∩ Fin) β†’ 𝑦 ∈ ran (π‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ↦ ran π‘₯)))
8180ssrdv 3955 . . . . . 6 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ (𝒫 𝐴 ∩ Fin) βŠ† ran (π‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ↦ ran π‘₯))
8238, 81eqssd 3966 . . . . 5 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ ran (π‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ↦ ran π‘₯) = (𝒫 𝐴 ∩ Fin))
83 df-fo 6507 . . . . 5 ((π‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ↦ ran π‘₯):βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛)–ontoβ†’(𝒫 𝐴 ∩ Fin) ↔ ((π‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ↦ ran π‘₯) Fn βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ∧ ran (π‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ↦ ran π‘₯) = (𝒫 𝐴 ∩ Fin)))
8437, 82, 83sylanbrc 584 . . . 4 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ (π‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ↦ ran π‘₯):βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛)–ontoβ†’(𝒫 𝐴 ∩ Fin))
85 fodomnum 10000 . . . 4 (βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ∈ dom card β†’ ((π‘₯ ∈ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ↦ ran π‘₯):βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛)–ontoβ†’(𝒫 𝐴 ∩ Fin) β†’ (𝒫 𝐴 ∩ Fin) β‰Ό βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛)))
8612, 84, 85sylc 65 . . 3 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ (𝒫 𝐴 ∩ Fin) β‰Ό βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛))
87 domtr 8954 . . 3 (((𝒫 𝐴 ∩ Fin) β‰Ό βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) ∧ βˆͺ 𝑛 ∈ Ο‰ (𝐴 ↑m 𝑛) β‰Ό 𝐴) β†’ (𝒫 𝐴 ∩ Fin) β‰Ό 𝐴)
8886, 10, 87syl2anc 585 . 2 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ (𝒫 𝐴 ∩ Fin) β‰Ό 𝐴)
89 pwexg 5338 . . . . 5 (𝐴 ∈ dom card β†’ 𝒫 𝐴 ∈ V)
9089adantr 482 . . . 4 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ 𝒫 𝐴 ∈ V)
91 inex1g 5281 . . . 4 (𝒫 𝐴 ∈ V β†’ (𝒫 𝐴 ∩ Fin) ∈ V)
9290, 91syl 17 . . 3 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ (𝒫 𝐴 ∩ Fin) ∈ V)
93 infpwfidom 9971 . . 3 ((𝒫 𝐴 ∩ Fin) ∈ V β†’ 𝐴 β‰Ό (𝒫 𝐴 ∩ Fin))
9492, 93syl 17 . 2 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ 𝐴 β‰Ό (𝒫 𝐴 ∩ Fin))
95 sbth 9044 . 2 (((𝒫 𝐴 ∩ Fin) β‰Ό 𝐴 ∧ 𝐴 β‰Ό (𝒫 𝐴 ∩ Fin)) β†’ (𝒫 𝐴 ∩ Fin) β‰ˆ 𝐴)
9688, 94, 95syl2anc 585 1 ((𝐴 ∈ dom card ∧ Ο‰ β‰Ό 𝐴) β†’ (𝒫 𝐴 ∩ Fin) β‰ˆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107   β‰  wne 2944  βˆƒwrex 3074  Vcvv 3448   ∩ cin 3914   βŠ† wss 3915  βˆ…c0 4287  π’« cpw 4565  βˆͺ ciun 4959   class class class wbr 5110   ↦ cmpt 5193   Γ— cxp 5636  dom cdm 5638  ran crn 5639   Fn wfn 6496  βŸΆwf 6497  β€“ontoβ†’wfo 6499  β€“1-1-ontoβ†’wf1o 6500  (class class class)co 7362  Ο‰com 7807   ↑m cmap 8772   β‰ˆ cen 8887   β‰Ό cdom 8888  Fincfn 8890  cardccrd 9878
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-seqom 8399  df-1o 8417  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-oi 9453  df-card 9882  df-acn 9885
This theorem is referenced by:  inffien  10006  isnumbasgrplem3  41461
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