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Theorem elrfi 38720
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 3428 . . 3 (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) → 𝐴 ∈ V)
21a1i 11 . 2 ((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) → 𝐴 ∈ V))
3 inex1g 5077 . . . . 5 (𝐵𝑉 → (𝐵 𝑣) ∈ V)
4 eleq1 2848 . . . . 5 (𝐴 = (𝐵 𝑣) → (𝐴 ∈ V ↔ (𝐵 𝑣) ∈ V))
53, 4syl5ibrcom 239 . . . 4 (𝐵𝑉 → (𝐴 = (𝐵 𝑣) → 𝐴 ∈ V))
65rexlimdvw 3230 . . 3 (𝐵𝑉 → (∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣) → 𝐴 ∈ V))
76adantr 473 . 2 ((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → (∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣) → 𝐴 ∈ V))
8 simpr 477 . . . . 5 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → 𝐴 ∈ V)
9 snex 5185 . . . . . 6 {𝐵} ∈ V
10 pwexg 5129 . . . . . . . 8 (𝐵𝑉 → 𝒫 𝐵 ∈ V)
1110ad2antrr 714 . . . . . . 7 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → 𝒫 𝐵 ∈ V)
12 simplr 757 . . . . . . 7 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → 𝐶 ⊆ 𝒫 𝐵)
1311, 12ssexd 5081 . . . . . 6 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → 𝐶 ∈ V)
14 unexg 7288 . . . . . 6 (({𝐵} ∈ V ∧ 𝐶 ∈ V) → ({𝐵} ∪ 𝐶) ∈ V)
159, 13, 14sylancr 579 . . . . 5 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → ({𝐵} ∪ 𝐶) ∈ V)
16 elfi 8671 . . . . 5 ((𝐴 ∈ V ∧ ({𝐵} ∪ 𝐶) ∈ V) → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) ↔ ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤))
178, 15, 16syl2anc 576 . . . 4 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) ↔ ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤))
18 inss1 4087 . . . . . . . . . . . 12 (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ⊆ 𝒫 ({𝐵} ∪ 𝐶)
19 uncom 4013 . . . . . . . . . . . . 13 ({𝐵} ∪ 𝐶) = (𝐶 ∪ {𝐵})
2019pweqi 4421 . . . . . . . . . . . 12 𝒫 ({𝐵} ∪ 𝐶) = 𝒫 (𝐶 ∪ {𝐵})
2118, 20sseqtri 3888 . . . . . . . . . . 11 (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ⊆ 𝒫 (𝐶 ∪ {𝐵})
2221sseli 3849 . . . . . . . . . 10 (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) → 𝑤 ∈ 𝒫 (𝐶 ∪ {𝐵}))
239elpwun 7307 . . . . . . . . . 10 (𝑤 ∈ 𝒫 (𝐶 ∪ {𝐵}) ↔ (𝑤 ∖ {𝐵}) ∈ 𝒫 𝐶)
2422, 23sylib 210 . . . . . . . . 9 (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) → (𝑤 ∖ {𝐵}) ∈ 𝒫 𝐶)
2524ad2antrl 716 . . . . . . . 8 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → (𝑤 ∖ {𝐵}) ∈ 𝒫 𝐶)
26 inss2 4088 . . . . . . . . . . 11 (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ⊆ Fin
2726sseli 3849 . . . . . . . . . 10 (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) → 𝑤 ∈ Fin)
28 diffi 8544 . . . . . . . . . 10 (𝑤 ∈ Fin → (𝑤 ∖ {𝐵}) ∈ Fin)
2927, 28syl 17 . . . . . . . . 9 (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) → (𝑤 ∖ {𝐵}) ∈ Fin)
3029ad2antrl 716 . . . . . . . 8 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → (𝑤 ∖ {𝐵}) ∈ Fin)
3125, 30elind 4054 . . . . . . 7 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → (𝑤 ∖ {𝐵}) ∈ (𝒫 𝐶 ∩ Fin))
32 incom 4061 . . . . . . . . . . . 12 (𝐵𝐴) = (𝐴𝐵)
33 simprr 761 . . . . . . . . . . . . . 14 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 = 𝑤)
34 simplr 757 . . . . . . . . . . . . . . . . . 18 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 ∈ V)
3533, 34eqeltrrd 2862 . . . . . . . . . . . . . . . . 17 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤 ∈ V)
36 intex 5093 . . . . . . . . . . . . . . . . 17 (𝑤 ≠ ∅ ↔ 𝑤 ∈ V)
3735, 36sylibr 226 . . . . . . . . . . . . . . . 16 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤 ≠ ∅)
38 intssuni 4768 . . . . . . . . . . . . . . . 16 (𝑤 ≠ ∅ → 𝑤 𝑤)
3937, 38syl 17 . . . . . . . . . . . . . . 15 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤 𝑤)
4018sseli 3849 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) → 𝑤 ∈ 𝒫 ({𝐵} ∪ 𝐶))
4140elpwid 4429 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) → 𝑤 ⊆ ({𝐵} ∪ 𝐶))
4241ad2antrl 716 . . . . . . . . . . . . . . . . 17 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤 ⊆ ({𝐵} ∪ 𝐶))
43 pwidg 4432 . . . . . . . . . . . . . . . . . . . . 21 (𝐵𝑉𝐵 ∈ 𝒫 𝐵)
4443snssd 4613 . . . . . . . . . . . . . . . . . . . 20 (𝐵𝑉 → {𝐵} ⊆ 𝒫 𝐵)
4544adantr 473 . . . . . . . . . . . . . . . . . . 19 ((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → {𝐵} ⊆ 𝒫 𝐵)
46 simpr 477 . . . . . . . . . . . . . . . . . . 19 ((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → 𝐶 ⊆ 𝒫 𝐵)
4745, 46unssd 4045 . . . . . . . . . . . . . . . . . 18 ((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → ({𝐵} ∪ 𝐶) ⊆ 𝒫 𝐵)
4847ad2antrr 714 . . . . . . . . . . . . . . . . 17 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → ({𝐵} ∪ 𝐶) ⊆ 𝒫 𝐵)
4942, 48sstrd 3863 . . . . . . . . . . . . . . . 16 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤 ⊆ 𝒫 𝐵)
50 sspwuni 4885 . . . . . . . . . . . . . . . 16 (𝑤 ⊆ 𝒫 𝐵 𝑤𝐵)
5149, 50sylib 210 . . . . . . . . . . . . . . 15 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤𝐵)
5239, 51sstrd 3863 . . . . . . . . . . . . . 14 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤𝐵)
5333, 52eqsstrd 3890 . . . . . . . . . . . . 13 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴𝐵)
54 df-ss 3838 . . . . . . . . . . . . 13 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
5553, 54sylib 210 . . . . . . . . . . . 12 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → (𝐴𝐵) = 𝐴)
5632, 55syl5req 2822 . . . . . . . . . . 11 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 = (𝐵𝐴))
57 ineq2 4065 . . . . . . . . . . . 12 (𝐴 = 𝑤 → (𝐵𝐴) = (𝐵 𝑤))
5857ad2antll 717 . . . . . . . . . . 11 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → (𝐵𝐴) = (𝐵 𝑤))
5956, 58eqtrd 2809 . . . . . . . . . 10 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 = (𝐵 𝑤))
60 intun 4778 . . . . . . . . . . . 12 ({𝐵} ∪ 𝑤) = ( {𝐵} ∩ 𝑤)
61 intsng 4781 . . . . . . . . . . . . 13 (𝐵𝑉 {𝐵} = 𝐵)
6261ineq1d 4070 . . . . . . . . . . . 12 (𝐵𝑉 → ( {𝐵} ∩ 𝑤) = (𝐵 𝑤))
6360, 62syl5req 2822 . . . . . . . . . . 11 (𝐵𝑉 → (𝐵 𝑤) = ({𝐵} ∪ 𝑤))
6463ad3antrrr 718 . . . . . . . . . 10 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → (𝐵 𝑤) = ({𝐵} ∪ 𝑤))
6559, 64eqtrd 2809 . . . . . . . . 9 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 = ({𝐵} ∪ 𝑤))
66 undif2 4303 . . . . . . . . . 10 ({𝐵} ∪ (𝑤 ∖ {𝐵})) = ({𝐵} ∪ 𝑤)
6766inteqi 4750 . . . . . . . . 9 ({𝐵} ∪ (𝑤 ∖ {𝐵})) = ({𝐵} ∪ 𝑤)
6865, 67syl6eqr 2827 . . . . . . . 8 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 = ({𝐵} ∪ (𝑤 ∖ {𝐵})))
69 intun 4778 . . . . . . . . . 10 ({𝐵} ∪ (𝑤 ∖ {𝐵})) = ( {𝐵} ∩ (𝑤 ∖ {𝐵}))
7061ineq1d 4070 . . . . . . . . . 10 (𝐵𝑉 → ( {𝐵} ∩ (𝑤 ∖ {𝐵})) = (𝐵 (𝑤 ∖ {𝐵})))
7169, 70syl5eq 2821 . . . . . . . . 9 (𝐵𝑉 ({𝐵} ∪ (𝑤 ∖ {𝐵})) = (𝐵 (𝑤 ∖ {𝐵})))
7271ad3antrrr 718 . . . . . . . 8 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → ({𝐵} ∪ (𝑤 ∖ {𝐵})) = (𝐵 (𝑤 ∖ {𝐵})))
7368, 72eqtrd 2809 . . . . . . 7 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 = (𝐵 (𝑤 ∖ {𝐵})))
74 inteq 4749 . . . . . . . . 9 (𝑣 = (𝑤 ∖ {𝐵}) → 𝑣 = (𝑤 ∖ {𝐵}))
7574ineq2d 4071 . . . . . . . 8 (𝑣 = (𝑤 ∖ {𝐵}) → (𝐵 𝑣) = (𝐵 (𝑤 ∖ {𝐵})))
7675rspceeqv 3548 . . . . . . 7 (((𝑤 ∖ {𝐵}) ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝐴 = (𝐵 (𝑤 ∖ {𝐵}))) → ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣))
7731, 73, 76syl2anc 576 . . . . . 6 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣))
7877rexlimdvaa 3225 . . . . 5 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → (∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤 → ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣)))
79 ssun1 4032 . . . . . . . . . . . 12 {𝐵} ⊆ ({𝐵} ∪ 𝐶)
8079a1i 11 . . . . . . . . . . 11 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → {𝐵} ⊆ ({𝐵} ∪ 𝐶))
81 inss1 4087 . . . . . . . . . . . . . 14 (𝒫 𝐶 ∩ Fin) ⊆ 𝒫 𝐶
8281sseli 3849 . . . . . . . . . . . . 13 (𝑣 ∈ (𝒫 𝐶 ∩ Fin) → 𝑣 ∈ 𝒫 𝐶)
83 elpwi 4427 . . . . . . . . . . . . 13 (𝑣 ∈ 𝒫 𝐶𝑣𝐶)
84 ssun4 4035 . . . . . . . . . . . . 13 (𝑣𝐶𝑣 ⊆ ({𝐵} ∪ 𝐶))
8582, 83, 843syl 18 . . . . . . . . . . . 12 (𝑣 ∈ (𝒫 𝐶 ∩ Fin) → 𝑣 ⊆ ({𝐵} ∪ 𝐶))
8685adantl 474 . . . . . . . . . . 11 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑣 ⊆ ({𝐵} ∪ 𝐶))
8780, 86unssd 4045 . . . . . . . . . 10 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → ({𝐵} ∪ 𝑣) ⊆ ({𝐵} ∪ 𝐶))
88 vex 3413 . . . . . . . . . . . 12 𝑣 ∈ V
899, 88unex 7285 . . . . . . . . . . 11 ({𝐵} ∪ 𝑣) ∈ V
9089elpw 4423 . . . . . . . . . 10 (({𝐵} ∪ 𝑣) ∈ 𝒫 ({𝐵} ∪ 𝐶) ↔ ({𝐵} ∪ 𝑣) ⊆ ({𝐵} ∪ 𝐶))
9187, 90sylibr 226 . . . . . . . . 9 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → ({𝐵} ∪ 𝑣) ∈ 𝒫 ({𝐵} ∪ 𝐶))
92 snfi 8390 . . . . . . . . . 10 {𝐵} ∈ Fin
93 inss2 4088 . . . . . . . . . . . 12 (𝒫 𝐶 ∩ Fin) ⊆ Fin
9493sseli 3849 . . . . . . . . . . 11 (𝑣 ∈ (𝒫 𝐶 ∩ Fin) → 𝑣 ∈ Fin)
9594adantl 474 . . . . . . . . . 10 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑣 ∈ Fin)
96 unfi 8579 . . . . . . . . . 10 (({𝐵} ∈ Fin ∧ 𝑣 ∈ Fin) → ({𝐵} ∪ 𝑣) ∈ Fin)
9792, 95, 96sylancr 579 . . . . . . . . 9 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → ({𝐵} ∪ 𝑣) ∈ Fin)
9891, 97elind 4054 . . . . . . . 8 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → ({𝐵} ∪ 𝑣) ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin))
9961eqcomd 2779 . . . . . . . . . . 11 (𝐵𝑉𝐵 = {𝐵})
10099ineq1d 4070 . . . . . . . . . 10 (𝐵𝑉 → (𝐵 𝑣) = ( {𝐵} ∩ 𝑣))
101 intun 4778 . . . . . . . . . 10 ({𝐵} ∪ 𝑣) = ( {𝐵} ∩ 𝑣)
102100, 101syl6eqr 2827 . . . . . . . . 9 (𝐵𝑉 → (𝐵 𝑣) = ({𝐵} ∪ 𝑣))
103102ad3antrrr 718 . . . . . . . 8 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐵 𝑣) = ({𝐵} ∪ 𝑣))
104 inteq 4749 . . . . . . . . 9 (𝑤 = ({𝐵} ∪ 𝑣) → 𝑤 = ({𝐵} ∪ 𝑣))
105104rspceeqv 3548 . . . . . . . 8 ((({𝐵} ∪ 𝑣) ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ (𝐵 𝑣) = ({𝐵} ∪ 𝑣)) → ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)(𝐵 𝑣) = 𝑤)
10698, 103, 105syl2anc 576 . . . . . . 7 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)(𝐵 𝑣) = 𝑤)
107 eqeq1 2777 . . . . . . . 8 (𝐴 = (𝐵 𝑣) → (𝐴 = 𝑤 ↔ (𝐵 𝑣) = 𝑤))
108107rexbidv 3237 . . . . . . 7 (𝐴 = (𝐵 𝑣) → (∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤 ↔ ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)(𝐵 𝑣) = 𝑤))
109106, 108syl5ibrcom 239 . . . . . 6 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐴 = (𝐵 𝑣) → ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤))
110109rexlimdva 3224 . . . . 5 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → (∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣) → ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤))
11178, 110impbid 204 . . . 4 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → (∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤 ↔ ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣)))
11217, 111bitrd 271 . . 3 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) ↔ ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣)))
113112ex 405 . 2 ((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → (𝐴 ∈ V → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) ↔ ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣))))
1142, 7, 113pm5.21ndd 372 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 198  wa 387   = wceq 1508  wcel 2051  wne 2962  wrex 3084  Vcvv 3410  cdif 3821  cun 3822  cin 3823  wss 3824  c0 4173  𝒫 cpw 4417  {csn 4436   cuni 4709   cint 4746  cfv 6186  Fincfn 8305  ficfi 8668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-ral 3088  df-rex 3089  df-reu 3090  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-pss 3840  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-tp 4441  df-op 4443  df-uni 4710  df-int 4747  df-iun 4791  df-br 4927  df-opab 4989  df-mpt 5006  df-tr 5028  df-id 5309  df-eprel 5314  df-po 5323  df-so 5324  df-fr 5363  df-we 5365  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-pred 5984  df-ord 6030  df-on 6031  df-lim 6032  df-suc 6033  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-ov 6978  df-oprab 6979  df-mpo 6980  df-om 7396  df-wrecs 7749  df-recs 7811  df-rdg 7849  df-1o 7904  df-oadd 7908  df-er 8088  df-en 8306  df-fin 8309  df-fi 8669
This theorem is referenced by:  elrfirn  38721
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