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Theorem elrfi 42351
Description: Elementhood in a set of relative finite intersections. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
elrfi ((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) ↔ ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣)))
Distinct variable groups:   𝑣,𝐴   𝑣,𝐵   𝑣,𝐶   𝑣,𝑉

Proof of Theorem elrfi
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 elex 3482 . . 3 (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) → 𝐴 ∈ V)
21a1i 11 . 2 ((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) → 𝐴 ∈ V))
3 inex1g 5324 . . . . 5 (𝐵𝑉 → (𝐵 𝑣) ∈ V)
4 eleq1 2814 . . . . 5 (𝐴 = (𝐵 𝑣) → (𝐴 ∈ V ↔ (𝐵 𝑣) ∈ V))
53, 4syl5ibrcom 246 . . . 4 (𝐵𝑉 → (𝐴 = (𝐵 𝑣) → 𝐴 ∈ V))
65rexlimdvw 3150 . . 3 (𝐵𝑉 → (∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣) → 𝐴 ∈ V))
76adantr 479 . 2 ((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → (∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣) → 𝐴 ∈ V))
8 simpr 483 . . . . 5 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → 𝐴 ∈ V)
9 snex 5437 . . . . . 6 {𝐵} ∈ V
10 pwexg 5382 . . . . . . . 8 (𝐵𝑉 → 𝒫 𝐵 ∈ V)
1110ad2antrr 724 . . . . . . 7 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → 𝒫 𝐵 ∈ V)
12 simplr 767 . . . . . . 7 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → 𝐶 ⊆ 𝒫 𝐵)
1311, 12ssexd 5329 . . . . . 6 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → 𝐶 ∈ V)
14 unexg 7757 . . . . . 6 (({𝐵} ∈ V ∧ 𝐶 ∈ V) → ({𝐵} ∪ 𝐶) ∈ V)
159, 13, 14sylancr 585 . . . . 5 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → ({𝐵} ∪ 𝐶) ∈ V)
16 elfi 9456 . . . . 5 ((𝐴 ∈ V ∧ ({𝐵} ∪ 𝐶) ∈ V) → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) ↔ ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤))
178, 15, 16syl2anc 582 . . . 4 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) ↔ ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤))
18 inss1 4230 . . . . . . . . . . . 12 (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ⊆ 𝒫 ({𝐵} ∪ 𝐶)
19 uncom 4153 . . . . . . . . . . . . 13 ({𝐵} ∪ 𝐶) = (𝐶 ∪ {𝐵})
2019pweqi 4623 . . . . . . . . . . . 12 𝒫 ({𝐵} ∪ 𝐶) = 𝒫 (𝐶 ∪ {𝐵})
2118, 20sseqtri 4016 . . . . . . . . . . 11 (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ⊆ 𝒫 (𝐶 ∪ {𝐵})
2221sseli 3975 . . . . . . . . . 10 (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) → 𝑤 ∈ 𝒫 (𝐶 ∪ {𝐵}))
239elpwun 7777 . . . . . . . . . 10 (𝑤 ∈ 𝒫 (𝐶 ∪ {𝐵}) ↔ (𝑤 ∖ {𝐵}) ∈ 𝒫 𝐶)
2422, 23sylib 217 . . . . . . . . 9 (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) → (𝑤 ∖ {𝐵}) ∈ 𝒫 𝐶)
2524ad2antrl 726 . . . . . . . 8 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → (𝑤 ∖ {𝐵}) ∈ 𝒫 𝐶)
26 inss2 4231 . . . . . . . . . . 11 (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ⊆ Fin
2726sseli 3975 . . . . . . . . . 10 (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) → 𝑤 ∈ Fin)
28 diffi 9213 . . . . . . . . . 10 (𝑤 ∈ Fin → (𝑤 ∖ {𝐵}) ∈ Fin)
2927, 28syl 17 . . . . . . . . 9 (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) → (𝑤 ∖ {𝐵}) ∈ Fin)
3029ad2antrl 726 . . . . . . . 8 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → (𝑤 ∖ {𝐵}) ∈ Fin)
3125, 30elind 4195 . . . . . . 7 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → (𝑤 ∖ {𝐵}) ∈ (𝒫 𝐶 ∩ Fin))
32 incom 4202 . . . . . . . . . . . 12 (𝐵𝐴) = (𝐴𝐵)
33 simprr 771 . . . . . . . . . . . . . 14 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 = 𝑤)
34 simplr 767 . . . . . . . . . . . . . . . . . 18 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 ∈ V)
3533, 34eqeltrrd 2827 . . . . . . . . . . . . . . . . 17 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤 ∈ V)
36 intex 5344 . . . . . . . . . . . . . . . . 17 (𝑤 ≠ ∅ ↔ 𝑤 ∈ V)
3735, 36sylibr 233 . . . . . . . . . . . . . . . 16 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤 ≠ ∅)
38 intssuni 4978 . . . . . . . . . . . . . . . 16 (𝑤 ≠ ∅ → 𝑤 𝑤)
3937, 38syl 17 . . . . . . . . . . . . . . 15 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤 𝑤)
4018sseli 3975 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) → 𝑤 ∈ 𝒫 ({𝐵} ∪ 𝐶))
4140elpwid 4616 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) → 𝑤 ⊆ ({𝐵} ∪ 𝐶))
4241ad2antrl 726 . . . . . . . . . . . . . . . . 17 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤 ⊆ ({𝐵} ∪ 𝐶))
43 pwidg 4627 . . . . . . . . . . . . . . . . . . . . 21 (𝐵𝑉𝐵 ∈ 𝒫 𝐵)
4443snssd 4818 . . . . . . . . . . . . . . . . . . . 20 (𝐵𝑉 → {𝐵} ⊆ 𝒫 𝐵)
4544adantr 479 . . . . . . . . . . . . . . . . . . 19 ((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → {𝐵} ⊆ 𝒫 𝐵)
46 simpr 483 . . . . . . . . . . . . . . . . . . 19 ((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → 𝐶 ⊆ 𝒫 𝐵)
4745, 46unssd 4187 . . . . . . . . . . . . . . . . . 18 ((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → ({𝐵} ∪ 𝐶) ⊆ 𝒫 𝐵)
4847ad2antrr 724 . . . . . . . . . . . . . . . . 17 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → ({𝐵} ∪ 𝐶) ⊆ 𝒫 𝐵)
4942, 48sstrd 3990 . . . . . . . . . . . . . . . 16 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤 ⊆ 𝒫 𝐵)
50 sspwuni 5108 . . . . . . . . . . . . . . . 16 (𝑤 ⊆ 𝒫 𝐵 𝑤𝐵)
5149, 50sylib 217 . . . . . . . . . . . . . . 15 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤𝐵)
5239, 51sstrd 3990 . . . . . . . . . . . . . 14 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤𝐵)
5333, 52eqsstrd 4018 . . . . . . . . . . . . 13 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴𝐵)
54 dfss2 3965 . . . . . . . . . . . . 13 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
5553, 54sylib 217 . . . . . . . . . . . 12 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → (𝐴𝐵) = 𝐴)
5632, 55eqtr2id 2779 . . . . . . . . . . 11 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 = (𝐵𝐴))
57 ineq2 4207 . . . . . . . . . . . 12 (𝐴 = 𝑤 → (𝐵𝐴) = (𝐵 𝑤))
5857ad2antll 727 . . . . . . . . . . 11 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → (𝐵𝐴) = (𝐵 𝑤))
5956, 58eqtrd 2766 . . . . . . . . . 10 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 = (𝐵 𝑤))
60 intun 4988 . . . . . . . . . . . 12 ({𝐵} ∪ 𝑤) = ( {𝐵} ∩ 𝑤)
61 intsng 4993 . . . . . . . . . . . . 13 (𝐵𝑉 {𝐵} = 𝐵)
6261ineq1d 4212 . . . . . . . . . . . 12 (𝐵𝑉 → ( {𝐵} ∩ 𝑤) = (𝐵 𝑤))
6360, 62eqtr2id 2779 . . . . . . . . . . 11 (𝐵𝑉 → (𝐵 𝑤) = ({𝐵} ∪ 𝑤))
6463ad3antrrr 728 . . . . . . . . . 10 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → (𝐵 𝑤) = ({𝐵} ∪ 𝑤))
6559, 64eqtrd 2766 . . . . . . . . 9 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 = ({𝐵} ∪ 𝑤))
66 undif2 4481 . . . . . . . . . 10 ({𝐵} ∪ (𝑤 ∖ {𝐵})) = ({𝐵} ∪ 𝑤)
6766inteqi 4958 . . . . . . . . 9 ({𝐵} ∪ (𝑤 ∖ {𝐵})) = ({𝐵} ∪ 𝑤)
6865, 67eqtr4di 2784 . . . . . . . 8 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 = ({𝐵} ∪ (𝑤 ∖ {𝐵})))
69 intun 4988 . . . . . . . . . 10 ({𝐵} ∪ (𝑤 ∖ {𝐵})) = ( {𝐵} ∩ (𝑤 ∖ {𝐵}))
7061ineq1d 4212 . . . . . . . . . 10 (𝐵𝑉 → ( {𝐵} ∩ (𝑤 ∖ {𝐵})) = (𝐵 (𝑤 ∖ {𝐵})))
7169, 70eqtrid 2778 . . . . . . . . 9 (𝐵𝑉 ({𝐵} ∪ (𝑤 ∖ {𝐵})) = (𝐵 (𝑤 ∖ {𝐵})))
7271ad3antrrr 728 . . . . . . . 8 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → ({𝐵} ∪ (𝑤 ∖ {𝐵})) = (𝐵 (𝑤 ∖ {𝐵})))
7368, 72eqtrd 2766 . . . . . . 7 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 = (𝐵 (𝑤 ∖ {𝐵})))
74 inteq 4957 . . . . . . . . 9 (𝑣 = (𝑤 ∖ {𝐵}) → 𝑣 = (𝑤 ∖ {𝐵}))
7574ineq2d 4213 . . . . . . . 8 (𝑣 = (𝑤 ∖ {𝐵}) → (𝐵 𝑣) = (𝐵 (𝑤 ∖ {𝐵})))
7675rspceeqv 3630 . . . . . . 7 (((𝑤 ∖ {𝐵}) ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝐴 = (𝐵 (𝑤 ∖ {𝐵}))) → ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣))
7731, 73, 76syl2anc 582 . . . . . 6 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣))
7877rexlimdvaa 3146 . . . . 5 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → (∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤 → ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣)))
79 ssun1 4173 . . . . . . . . . . . 12 {𝐵} ⊆ ({𝐵} ∪ 𝐶)
8079a1i 11 . . . . . . . . . . 11 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → {𝐵} ⊆ ({𝐵} ∪ 𝐶))
81 inss1 4230 . . . . . . . . . . . . . 14 (𝒫 𝐶 ∩ Fin) ⊆ 𝒫 𝐶
8281sseli 3975 . . . . . . . . . . . . 13 (𝑣 ∈ (𝒫 𝐶 ∩ Fin) → 𝑣 ∈ 𝒫 𝐶)
83 elpwi 4614 . . . . . . . . . . . . 13 (𝑣 ∈ 𝒫 𝐶𝑣𝐶)
84 ssun4 4176 . . . . . . . . . . . . 13 (𝑣𝐶𝑣 ⊆ ({𝐵} ∪ 𝐶))
8582, 83, 843syl 18 . . . . . . . . . . . 12 (𝑣 ∈ (𝒫 𝐶 ∩ Fin) → 𝑣 ⊆ ({𝐵} ∪ 𝐶))
8685adantl 480 . . . . . . . . . . 11 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑣 ⊆ ({𝐵} ∪ 𝐶))
8780, 86unssd 4187 . . . . . . . . . 10 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → ({𝐵} ∪ 𝑣) ⊆ ({𝐵} ∪ 𝐶))
88 vex 3466 . . . . . . . . . . . 12 𝑣 ∈ V
899, 88unex 7754 . . . . . . . . . . 11 ({𝐵} ∪ 𝑣) ∈ V
9089elpw 4611 . . . . . . . . . 10 (({𝐵} ∪ 𝑣) ∈ 𝒫 ({𝐵} ∪ 𝐶) ↔ ({𝐵} ∪ 𝑣) ⊆ ({𝐵} ∪ 𝐶))
9187, 90sylibr 233 . . . . . . . . 9 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → ({𝐵} ∪ 𝑣) ∈ 𝒫 ({𝐵} ∪ 𝐶))
92 snfi 9081 . . . . . . . . . 10 {𝐵} ∈ Fin
93 inss2 4231 . . . . . . . . . . . 12 (𝒫 𝐶 ∩ Fin) ⊆ Fin
9493sseli 3975 . . . . . . . . . . 11 (𝑣 ∈ (𝒫 𝐶 ∩ Fin) → 𝑣 ∈ Fin)
9594adantl 480 . . . . . . . . . 10 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑣 ∈ Fin)
96 unfi 9210 . . . . . . . . . 10 (({𝐵} ∈ Fin ∧ 𝑣 ∈ Fin) → ({𝐵} ∪ 𝑣) ∈ Fin)
9792, 95, 96sylancr 585 . . . . . . . . 9 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → ({𝐵} ∪ 𝑣) ∈ Fin)
9891, 97elind 4195 . . . . . . . 8 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → ({𝐵} ∪ 𝑣) ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin))
9961eqcomd 2732 . . . . . . . . . . 11 (𝐵𝑉𝐵 = {𝐵})
10099ineq1d 4212 . . . . . . . . . 10 (𝐵𝑉 → (𝐵 𝑣) = ( {𝐵} ∩ 𝑣))
101 intun 4988 . . . . . . . . . 10 ({𝐵} ∪ 𝑣) = ( {𝐵} ∩ 𝑣)
102100, 101eqtr4di 2784 . . . . . . . . 9 (𝐵𝑉 → (𝐵 𝑣) = ({𝐵} ∪ 𝑣))
103102ad3antrrr 728 . . . . . . . 8 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐵 𝑣) = ({𝐵} ∪ 𝑣))
104 inteq 4957 . . . . . . . . 9 (𝑤 = ({𝐵} ∪ 𝑣) → 𝑤 = ({𝐵} ∪ 𝑣))
105104rspceeqv 3630 . . . . . . . 8 ((({𝐵} ∪ 𝑣) ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ (𝐵 𝑣) = ({𝐵} ∪ 𝑣)) → ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)(𝐵 𝑣) = 𝑤)
10698, 103, 105syl2anc 582 . . . . . . 7 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)(𝐵 𝑣) = 𝑤)
107 eqeq1 2730 . . . . . . . 8 (𝐴 = (𝐵 𝑣) → (𝐴 = 𝑤 ↔ (𝐵 𝑣) = 𝑤))
108107rexbidv 3169 . . . . . . 7 (𝐴 = (𝐵 𝑣) → (∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤 ↔ ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)(𝐵 𝑣) = 𝑤))
109106, 108syl5ibrcom 246 . . . . . 6 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐴 = (𝐵 𝑣) → ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤))
110109rexlimdva 3145 . . . . 5 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → (∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣) → ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤))
11178, 110impbid 211 . . . 4 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → (∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤 ↔ ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣)))
11217, 111bitrd 278 . . 3 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) ↔ ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣)))
113112ex 411 . 2 ((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → (𝐴 ∈ V → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) ↔ ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣))))
1142, 7, 113pm5.21ndd 378 1 ((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) ↔ ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wne 2930  wrex 3060  Vcvv 3462  cdif 3944  cun 3945  cin 3946  wss 3947  c0 4325  𝒫 cpw 4607  {csn 4633   cuni 4913   cint 4954  cfv 6554  Fincfn 8974  ficfi 9453
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 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-om 7877  df-1o 8496  df-en 8975  df-fin 8978  df-fi 9454
This theorem is referenced by:  elrfirn  42352
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