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Theorem elrfi 36141
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 3089 . . 3 (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) → 𝐴 ∈ V)
21a1i 11 . 2 ((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) → 𝐴 ∈ V))
3 inex1g 4628 . . . . 5 (𝐵𝑉 → (𝐵 𝑣) ∈ V)
4 eleq1 2580 . . . . 5 (𝐴 = (𝐵 𝑣) → (𝐴 ∈ V ↔ (𝐵 𝑣) ∈ V))
53, 4syl5ibrcom 235 . . . 4 (𝐵𝑉 → (𝐴 = (𝐵 𝑣) → 𝐴 ∈ V))
65rexlimdvw 2920 . . 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 475 . . . . 5 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → 𝐴 ∈ V)
9 snex 4734 . . . . . 6 {𝐵} ∈ V
10 pwexg 4675 . . . . . . . 8 (𝐵𝑉 → 𝒫 𝐵 ∈ V)
1110ad2antrr 757 . . . . . . 7 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → 𝒫 𝐵 ∈ V)
12 simplr 787 . . . . . . 7 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → 𝐶 ⊆ 𝒫 𝐵)
1311, 12ssexd 4632 . . . . . 6 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → 𝐶 ∈ V)
14 unexg 6731 . . . . . 6 (({𝐵} ∈ V ∧ 𝐶 ∈ V) → ({𝐵} ∪ 𝐶) ∈ V)
159, 13, 14sylancr 693 . . . . 5 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → ({𝐵} ∪ 𝐶) ∈ V)
16 elfi 8076 . . . . 5 ((𝐴 ∈ V ∧ ({𝐵} ∪ 𝐶) ∈ V) → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) ↔ ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤))
178, 15, 16syl2anc 690 . . . 4 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) ↔ ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤))
18 inss1 3698 . . . . . . . . . . . 12 (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ⊆ 𝒫 ({𝐵} ∪ 𝐶)
19 uncom 3623 . . . . . . . . . . . . 13 ({𝐵} ∪ 𝐶) = (𝐶 ∪ {𝐵})
2019pweqi 4015 . . . . . . . . . . . 12 𝒫 ({𝐵} ∪ 𝐶) = 𝒫 (𝐶 ∪ {𝐵})
2118, 20sseqtri 3504 . . . . . . . . . . 11 (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ⊆ 𝒫 (𝐶 ∪ {𝐵})
2221sseli 3468 . . . . . . . . . 10 (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) → 𝑤 ∈ 𝒫 (𝐶 ∪ {𝐵}))
239elpwun 6743 . . . . . . . . . 10 (𝑤 ∈ 𝒫 (𝐶 ∪ {𝐵}) ↔ (𝑤 ∖ {𝐵}) ∈ 𝒫 𝐶)
2422, 23sylib 206 . . . . . . . . 9 (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) → (𝑤 ∖ {𝐵}) ∈ 𝒫 𝐶)
2524ad2antrl 759 . . . . . . . 8 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → (𝑤 ∖ {𝐵}) ∈ 𝒫 𝐶)
26 inss2 3699 . . . . . . . . . . 11 (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ⊆ Fin
2726sseli 3468 . . . . . . . . . 10 (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) → 𝑤 ∈ Fin)
28 diffi 7951 . . . . . . . . . 10 (𝑤 ∈ Fin → (𝑤 ∖ {𝐵}) ∈ Fin)
2927, 28syl 17 . . . . . . . . 9 (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) → (𝑤 ∖ {𝐵}) ∈ Fin)
3029ad2antrl 759 . . . . . . . 8 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → (𝑤 ∖ {𝐵}) ∈ Fin)
3125, 30elind 3663 . . . . . . 7 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → (𝑤 ∖ {𝐵}) ∈ (𝒫 𝐶 ∩ Fin))
32 incom 3670 . . . . . . . . . . . 12 (𝐵𝐴) = (𝐴𝐵)
33 simprr 791 . . . . . . . . . . . . . 14 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 = 𝑤)
34 simplr 787 . . . . . . . . . . . . . . . . . 18 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 ∈ V)
3533, 34eqeltrrd 2593 . . . . . . . . . . . . . . . . 17 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤 ∈ V)
36 intex 4646 . . . . . . . . . . . . . . . . 17 (𝑤 ≠ ∅ ↔ 𝑤 ∈ V)
3735, 36sylibr 222 . . . . . . . . . . . . . . . 16 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤 ≠ ∅)
38 intssuni 4332 . . . . . . . . . . . . . . . 16 (𝑤 ≠ ∅ → 𝑤 𝑤)
3937, 38syl 17 . . . . . . . . . . . . . . 15 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤 𝑤)
4018sseli 3468 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) → 𝑤 ∈ 𝒫 ({𝐵} ∪ 𝐶))
4140elpwid 4021 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) → 𝑤 ⊆ ({𝐵} ∪ 𝐶))
4241ad2antrl 759 . . . . . . . . . . . . . . . . 17 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤 ⊆ ({𝐵} ∪ 𝐶))
43 pwidg 4024 . . . . . . . . . . . . . . . . . . . . 21 (𝐵𝑉𝐵 ∈ 𝒫 𝐵)
4443snssd 4184 . . . . . . . . . . . . . . . . . . . 20 (𝐵𝑉 → {𝐵} ⊆ 𝒫 𝐵)
4544adantr 479 . . . . . . . . . . . . . . . . . . 19 ((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → {𝐵} ⊆ 𝒫 𝐵)
46 simpr 475 . . . . . . . . . . . . . . . . . . 19 ((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → 𝐶 ⊆ 𝒫 𝐵)
4745, 46unssd 3655 . . . . . . . . . . . . . . . . . 18 ((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → ({𝐵} ∪ 𝐶) ⊆ 𝒫 𝐵)
4847ad2antrr 757 . . . . . . . . . . . . . . . . 17 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → ({𝐵} ∪ 𝐶) ⊆ 𝒫 𝐵)
4942, 48sstrd 3482 . . . . . . . . . . . . . . . 16 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤 ⊆ 𝒫 𝐵)
50 sspwuni 4445 . . . . . . . . . . . . . . . 16 (𝑤 ⊆ 𝒫 𝐵 𝑤𝐵)
5149, 50sylib 206 . . . . . . . . . . . . . . 15 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤𝐵)
5239, 51sstrd 3482 . . . . . . . . . . . . . 14 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤𝐵)
5333, 52eqsstrd 3506 . . . . . . . . . . . . 13 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴𝐵)
54 df-ss 3458 . . . . . . . . . . . . 13 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
5553, 54sylib 206 . . . . . . . . . . . 12 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → (𝐴𝐵) = 𝐴)
5632, 55syl5req 2561 . . . . . . . . . . 11 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 = (𝐵𝐴))
57 ineq2 3673 . . . . . . . . . . . 12 (𝐴 = 𝑤 → (𝐵𝐴) = (𝐵 𝑤))
5857ad2antll 760 . . . . . . . . . . 11 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → (𝐵𝐴) = (𝐵 𝑤))
5956, 58eqtrd 2548 . . . . . . . . . 10 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 = (𝐵 𝑤))
60 intun 4342 . . . . . . . . . . . 12 ({𝐵} ∪ 𝑤) = ( {𝐵} ∩ 𝑤)
61 intsng 4345 . . . . . . . . . . . . 13 (𝐵𝑉 {𝐵} = 𝐵)
6261ineq1d 3678 . . . . . . . . . . . 12 (𝐵𝑉 → ( {𝐵} ∩ 𝑤) = (𝐵 𝑤))
6360, 62syl5req 2561 . . . . . . . . . . 11 (𝐵𝑉 → (𝐵 𝑤) = ({𝐵} ∪ 𝑤))
6463ad3antrrr 761 . . . . . . . . . 10 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → (𝐵 𝑤) = ({𝐵} ∪ 𝑤))
6559, 64eqtrd 2548 . . . . . . . . 9 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 = ({𝐵} ∪ 𝑤))
66 undif2 3899 . . . . . . . . . 10 ({𝐵} ∪ (𝑤 ∖ {𝐵})) = ({𝐵} ∪ 𝑤)
6766inteqi 4312 . . . . . . . . 9 ({𝐵} ∪ (𝑤 ∖ {𝐵})) = ({𝐵} ∪ 𝑤)
6865, 67syl6eqr 2566 . . . . . . . 8 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 = ({𝐵} ∪ (𝑤 ∖ {𝐵})))
69 intun 4342 . . . . . . . . . 10 ({𝐵} ∪ (𝑤 ∖ {𝐵})) = ( {𝐵} ∩ (𝑤 ∖ {𝐵}))
7061ineq1d 3678 . . . . . . . . . 10 (𝐵𝑉 → ( {𝐵} ∩ (𝑤 ∖ {𝐵})) = (𝐵 (𝑤 ∖ {𝐵})))
7169, 70syl5eq 2560 . . . . . . . . 9 (𝐵𝑉 ({𝐵} ∪ (𝑤 ∖ {𝐵})) = (𝐵 (𝑤 ∖ {𝐵})))
7271ad3antrrr 761 . . . . . . . 8 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → ({𝐵} ∪ (𝑤 ∖ {𝐵})) = (𝐵 (𝑤 ∖ {𝐵})))
7368, 72eqtrd 2548 . . . . . . 7 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 = (𝐵 (𝑤 ∖ {𝐵})))
74 inteq 4311 . . . . . . . . . 10 (𝑣 = (𝑤 ∖ {𝐵}) → 𝑣 = (𝑤 ∖ {𝐵}))
7574ineq2d 3679 . . . . . . . . 9 (𝑣 = (𝑤 ∖ {𝐵}) → (𝐵 𝑣) = (𝐵 (𝑤 ∖ {𝐵})))
7675eqeq2d 2524 . . . . . . . 8 (𝑣 = (𝑤 ∖ {𝐵}) → (𝐴 = (𝐵 𝑣) ↔ 𝐴 = (𝐵 (𝑤 ∖ {𝐵}))))
7776rspcev 3186 . . . . . . 7 (((𝑤 ∖ {𝐵}) ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝐴 = (𝐵 (𝑤 ∖ {𝐵}))) → ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣))
7831, 73, 77syl2anc 690 . . . . . 6 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣))
7978rexlimdvaa 2918 . . . . 5 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → (∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤 → ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣)))
80 ssun1 3642 . . . . . . . . . . . 12 {𝐵} ⊆ ({𝐵} ∪ 𝐶)
8180a1i 11 . . . . . . . . . . 11 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → {𝐵} ⊆ ({𝐵} ∪ 𝐶))
82 inss1 3698 . . . . . . . . . . . . . 14 (𝒫 𝐶 ∩ Fin) ⊆ 𝒫 𝐶
8382sseli 3468 . . . . . . . . . . . . 13 (𝑣 ∈ (𝒫 𝐶 ∩ Fin) → 𝑣 ∈ 𝒫 𝐶)
84 elpwi 4020 . . . . . . . . . . . . 13 (𝑣 ∈ 𝒫 𝐶𝑣𝐶)
85 ssun4 3645 . . . . . . . . . . . . 13 (𝑣𝐶𝑣 ⊆ ({𝐵} ∪ 𝐶))
8683, 84, 853syl 18 . . . . . . . . . . . 12 (𝑣 ∈ (𝒫 𝐶 ∩ Fin) → 𝑣 ⊆ ({𝐵} ∪ 𝐶))
8786adantl 480 . . . . . . . . . . 11 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑣 ⊆ ({𝐵} ∪ 𝐶))
8881, 87unssd 3655 . . . . . . . . . 10 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → ({𝐵} ∪ 𝑣) ⊆ ({𝐵} ∪ 𝐶))
89 vex 3080 . . . . . . . . . . . 12 𝑣 ∈ V
909, 89unex 6728 . . . . . . . . . . 11 ({𝐵} ∪ 𝑣) ∈ V
9190elpw 4017 . . . . . . . . . 10 (({𝐵} ∪ 𝑣) ∈ 𝒫 ({𝐵} ∪ 𝐶) ↔ ({𝐵} ∪ 𝑣) ⊆ ({𝐵} ∪ 𝐶))
9288, 91sylibr 222 . . . . . . . . 9 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → ({𝐵} ∪ 𝑣) ∈ 𝒫 ({𝐵} ∪ 𝐶))
93 snfi 7797 . . . . . . . . . 10 {𝐵} ∈ Fin
94 inss2 3699 . . . . . . . . . . . 12 (𝒫 𝐶 ∩ Fin) ⊆ Fin
9594sseli 3468 . . . . . . . . . . 11 (𝑣 ∈ (𝒫 𝐶 ∩ Fin) → 𝑣 ∈ Fin)
9695adantl 480 . . . . . . . . . 10 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑣 ∈ Fin)
97 unfi 7986 . . . . . . . . . 10 (({𝐵} ∈ Fin ∧ 𝑣 ∈ Fin) → ({𝐵} ∪ 𝑣) ∈ Fin)
9893, 96, 97sylancr 693 . . . . . . . . 9 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → ({𝐵} ∪ 𝑣) ∈ Fin)
9992, 98elind 3663 . . . . . . . 8 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → ({𝐵} ∪ 𝑣) ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin))
10061eqcomd 2520 . . . . . . . . . . 11 (𝐵𝑉𝐵 = {𝐵})
101100ineq1d 3678 . . . . . . . . . 10 (𝐵𝑉 → (𝐵 𝑣) = ( {𝐵} ∩ 𝑣))
102 intun 4342 . . . . . . . . . 10 ({𝐵} ∪ 𝑣) = ( {𝐵} ∩ 𝑣)
103101, 102syl6eqr 2566 . . . . . . . . 9 (𝐵𝑉 → (𝐵 𝑣) = ({𝐵} ∪ 𝑣))
104103ad3antrrr 761 . . . . . . . 8 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐵 𝑣) = ({𝐵} ∪ 𝑣))
105 inteq 4311 . . . . . . . . . 10 (𝑤 = ({𝐵} ∪ 𝑣) → 𝑤 = ({𝐵} ∪ 𝑣))
106105eqeq2d 2524 . . . . . . . . 9 (𝑤 = ({𝐵} ∪ 𝑣) → ((𝐵 𝑣) = 𝑤 ↔ (𝐵 𝑣) = ({𝐵} ∪ 𝑣)))
107106rspcev 3186 . . . . . . . 8 ((({𝐵} ∪ 𝑣) ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ (𝐵 𝑣) = ({𝐵} ∪ 𝑣)) → ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)(𝐵 𝑣) = 𝑤)
10899, 104, 107syl2anc 690 . . . . . . 7 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)(𝐵 𝑣) = 𝑤)
109 eqeq1 2518 . . . . . . . 8 (𝐴 = (𝐵 𝑣) → (𝐴 = 𝑤 ↔ (𝐵 𝑣) = 𝑤))
110109rexbidv 2938 . . . . . . 7 (𝐴 = (𝐵 𝑣) → (∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤 ↔ ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)(𝐵 𝑣) = 𝑤))
111108, 110syl5ibrcom 235 . . . . . 6 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐴 = (𝐵 𝑣) → ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤))
112111rexlimdva 2917 . . . . 5 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → (∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣) → ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤))
11379, 112impbid 200 . . . 4 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → (∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤 ↔ ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣)))
11417, 113bitrd 266 . . 3 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) ↔ ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣)))
115114ex 448 . 2 ((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → (𝐴 ∈ V → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) ↔ ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣))))
1162, 7, 115pm5.21ndd 367 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 194  wa 382   = wceq 1474  wcel 1938  wne 2684  wrex 2801  Vcvv 3077  cdif 3441  cun 3442  cin 3443  wss 3444  c0 3777  𝒫 cpw 4011  {csn 4028   cuni 4270   cint 4308  cfv 5689  Fincfn 7715  ficfi 8073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6721
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-int 4309  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-ov 6428  df-oprab 6429  df-mpt2 6430  df-om 6832  df-wrecs 7167  df-recs 7229  df-rdg 7267  df-1o 7321  df-oadd 7325  df-er 7503  df-en 7716  df-fin 7719  df-fi 8074
This theorem is referenced by:  elrfirn  36142
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