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Theorem elrfi 39632
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 3462 . . 3 (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) → 𝐴 ∈ V)
21a1i 11 . 2 ((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) → 𝐴 ∈ V))
3 inex1g 5190 . . . . 5 (𝐵𝑉 → (𝐵 𝑣) ∈ V)
4 eleq1 2880 . . . . 5 (𝐴 = (𝐵 𝑣) → (𝐴 ∈ V ↔ (𝐵 𝑣) ∈ V))
53, 4syl5ibrcom 250 . . . 4 (𝐵𝑉 → (𝐴 = (𝐵 𝑣) → 𝐴 ∈ V))
65rexlimdvw 3252 . . 3 (𝐵𝑉 → (∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣) → 𝐴 ∈ V))
76adantr 484 . 2 ((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → (∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣) → 𝐴 ∈ V))
8 simpr 488 . . . . 5 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → 𝐴 ∈ V)
9 snex 5300 . . . . . 6 {𝐵} ∈ V
10 pwexg 5247 . . . . . . . 8 (𝐵𝑉 → 𝒫 𝐵 ∈ V)
1110ad2antrr 725 . . . . . . 7 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → 𝒫 𝐵 ∈ V)
12 simplr 768 . . . . . . 7 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → 𝐶 ⊆ 𝒫 𝐵)
1311, 12ssexd 5195 . . . . . 6 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → 𝐶 ∈ V)
14 unexg 7456 . . . . . 6 (({𝐵} ∈ V ∧ 𝐶 ∈ V) → ({𝐵} ∪ 𝐶) ∈ V)
159, 13, 14sylancr 590 . . . . 5 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → ({𝐵} ∪ 𝐶) ∈ V)
16 elfi 8865 . . . . 5 ((𝐴 ∈ V ∧ ({𝐵} ∪ 𝐶) ∈ V) → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) ↔ ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤))
178, 15, 16syl2anc 587 . . . 4 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) ↔ ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤))
18 inss1 4158 . . . . . . . . . . . 12 (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ⊆ 𝒫 ({𝐵} ∪ 𝐶)
19 uncom 4083 . . . . . . . . . . . . 13 ({𝐵} ∪ 𝐶) = (𝐶 ∪ {𝐵})
2019pweqi 4518 . . . . . . . . . . . 12 𝒫 ({𝐵} ∪ 𝐶) = 𝒫 (𝐶 ∪ {𝐵})
2118, 20sseqtri 3954 . . . . . . . . . . 11 (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ⊆ 𝒫 (𝐶 ∪ {𝐵})
2221sseli 3914 . . . . . . . . . 10 (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) → 𝑤 ∈ 𝒫 (𝐶 ∪ {𝐵}))
239elpwun 7475 . . . . . . . . . 10 (𝑤 ∈ 𝒫 (𝐶 ∪ {𝐵}) ↔ (𝑤 ∖ {𝐵}) ∈ 𝒫 𝐶)
2422, 23sylib 221 . . . . . . . . 9 (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) → (𝑤 ∖ {𝐵}) ∈ 𝒫 𝐶)
2524ad2antrl 727 . . . . . . . 8 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → (𝑤 ∖ {𝐵}) ∈ 𝒫 𝐶)
26 inss2 4159 . . . . . . . . . . 11 (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ⊆ Fin
2726sseli 3914 . . . . . . . . . 10 (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) → 𝑤 ∈ Fin)
28 diffi 8738 . . . . . . . . . 10 (𝑤 ∈ Fin → (𝑤 ∖ {𝐵}) ∈ Fin)
2927, 28syl 17 . . . . . . . . 9 (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) → (𝑤 ∖ {𝐵}) ∈ Fin)
3029ad2antrl 727 . . . . . . . 8 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → (𝑤 ∖ {𝐵}) ∈ Fin)
3125, 30elind 4124 . . . . . . 7 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → (𝑤 ∖ {𝐵}) ∈ (𝒫 𝐶 ∩ Fin))
32 incom 4131 . . . . . . . . . . . 12 (𝐵𝐴) = (𝐴𝐵)
33 simprr 772 . . . . . . . . . . . . . 14 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 = 𝑤)
34 simplr 768 . . . . . . . . . . . . . . . . . 18 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 ∈ V)
3533, 34eqeltrrd 2894 . . . . . . . . . . . . . . . . 17 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤 ∈ V)
36 intex 5207 . . . . . . . . . . . . . . . . 17 (𝑤 ≠ ∅ ↔ 𝑤 ∈ V)
3735, 36sylibr 237 . . . . . . . . . . . . . . . 16 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤 ≠ ∅)
38 intssuni 4863 . . . . . . . . . . . . . . . 16 (𝑤 ≠ ∅ → 𝑤 𝑤)
3937, 38syl 17 . . . . . . . . . . . . . . 15 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤 𝑤)
4018sseli 3914 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) → 𝑤 ∈ 𝒫 ({𝐵} ∪ 𝐶))
4140elpwid 4511 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) → 𝑤 ⊆ ({𝐵} ∪ 𝐶))
4241ad2antrl 727 . . . . . . . . . . . . . . . . 17 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤 ⊆ ({𝐵} ∪ 𝐶))
43 pwidg 4522 . . . . . . . . . . . . . . . . . . . . 21 (𝐵𝑉𝐵 ∈ 𝒫 𝐵)
4443snssd 4705 . . . . . . . . . . . . . . . . . . . 20 (𝐵𝑉 → {𝐵} ⊆ 𝒫 𝐵)
4544adantr 484 . . . . . . . . . . . . . . . . . . 19 ((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → {𝐵} ⊆ 𝒫 𝐵)
46 simpr 488 . . . . . . . . . . . . . . . . . . 19 ((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → 𝐶 ⊆ 𝒫 𝐵)
4745, 46unssd 4116 . . . . . . . . . . . . . . . . . 18 ((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → ({𝐵} ∪ 𝐶) ⊆ 𝒫 𝐵)
4847ad2antrr 725 . . . . . . . . . . . . . . . . 17 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → ({𝐵} ∪ 𝐶) ⊆ 𝒫 𝐵)
4942, 48sstrd 3928 . . . . . . . . . . . . . . . 16 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤 ⊆ 𝒫 𝐵)
50 sspwuni 4988 . . . . . . . . . . . . . . . 16 (𝑤 ⊆ 𝒫 𝐵 𝑤𝐵)
5149, 50sylib 221 . . . . . . . . . . . . . . 15 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤𝐵)
5239, 51sstrd 3928 . . . . . . . . . . . . . 14 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤𝐵)
5333, 52eqsstrd 3956 . . . . . . . . . . . . 13 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴𝐵)
54 df-ss 3901 . . . . . . . . . . . . 13 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
5553, 54sylib 221 . . . . . . . . . . . 12 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → (𝐴𝐵) = 𝐴)
5632, 55syl5req 2849 . . . . . . . . . . 11 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 = (𝐵𝐴))
57 ineq2 4136 . . . . . . . . . . . 12 (𝐴 = 𝑤 → (𝐵𝐴) = (𝐵 𝑤))
5857ad2antll 728 . . . . . . . . . . 11 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → (𝐵𝐴) = (𝐵 𝑤))
5956, 58eqtrd 2836 . . . . . . . . . 10 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 = (𝐵 𝑤))
60 intun 4873 . . . . . . . . . . . 12 ({𝐵} ∪ 𝑤) = ( {𝐵} ∩ 𝑤)
61 intsng 4876 . . . . . . . . . . . . 13 (𝐵𝑉 {𝐵} = 𝐵)
6261ineq1d 4141 . . . . . . . . . . . 12 (𝐵𝑉 → ( {𝐵} ∩ 𝑤) = (𝐵 𝑤))
6360, 62syl5req 2849 . . . . . . . . . . 11 (𝐵𝑉 → (𝐵 𝑤) = ({𝐵} ∪ 𝑤))
6463ad3antrrr 729 . . . . . . . . . 10 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → (𝐵 𝑤) = ({𝐵} ∪ 𝑤))
6559, 64eqtrd 2836 . . . . . . . . 9 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 = ({𝐵} ∪ 𝑤))
66 undif2 4386 . . . . . . . . . 10 ({𝐵} ∪ (𝑤 ∖ {𝐵})) = ({𝐵} ∪ 𝑤)
6766inteqi 4845 . . . . . . . . 9 ({𝐵} ∪ (𝑤 ∖ {𝐵})) = ({𝐵} ∪ 𝑤)
6865, 67eqtr4di 2854 . . . . . . . 8 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 = ({𝐵} ∪ (𝑤 ∖ {𝐵})))
69 intun 4873 . . . . . . . . . 10 ({𝐵} ∪ (𝑤 ∖ {𝐵})) = ( {𝐵} ∩ (𝑤 ∖ {𝐵}))
7061ineq1d 4141 . . . . . . . . . 10 (𝐵𝑉 → ( {𝐵} ∩ (𝑤 ∖ {𝐵})) = (𝐵 (𝑤 ∖ {𝐵})))
7169, 70syl5eq 2848 . . . . . . . . 9 (𝐵𝑉 ({𝐵} ∪ (𝑤 ∖ {𝐵})) = (𝐵 (𝑤 ∖ {𝐵})))
7271ad3antrrr 729 . . . . . . . 8 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → ({𝐵} ∪ (𝑤 ∖ {𝐵})) = (𝐵 (𝑤 ∖ {𝐵})))
7368, 72eqtrd 2836 . . . . . . 7 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 = (𝐵 (𝑤 ∖ {𝐵})))
74 inteq 4844 . . . . . . . . 9 (𝑣 = (𝑤 ∖ {𝐵}) → 𝑣 = (𝑤 ∖ {𝐵}))
7574ineq2d 4142 . . . . . . . 8 (𝑣 = (𝑤 ∖ {𝐵}) → (𝐵 𝑣) = (𝐵 (𝑤 ∖ {𝐵})))
7675rspceeqv 3589 . . . . . . 7 (((𝑤 ∖ {𝐵}) ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝐴 = (𝐵 (𝑤 ∖ {𝐵}))) → ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣))
7731, 73, 76syl2anc 587 . . . . . 6 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣))
7877rexlimdvaa 3247 . . . . 5 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → (∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤 → ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣)))
79 ssun1 4102 . . . . . . . . . . . 12 {𝐵} ⊆ ({𝐵} ∪ 𝐶)
8079a1i 11 . . . . . . . . . . 11 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → {𝐵} ⊆ ({𝐵} ∪ 𝐶))
81 inss1 4158 . . . . . . . . . . . . . 14 (𝒫 𝐶 ∩ Fin) ⊆ 𝒫 𝐶
8281sseli 3914 . . . . . . . . . . . . 13 (𝑣 ∈ (𝒫 𝐶 ∩ Fin) → 𝑣 ∈ 𝒫 𝐶)
83 elpwi 4509 . . . . . . . . . . . . 13 (𝑣 ∈ 𝒫 𝐶𝑣𝐶)
84 ssun4 4105 . . . . . . . . . . . . 13 (𝑣𝐶𝑣 ⊆ ({𝐵} ∪ 𝐶))
8582, 83, 843syl 18 . . . . . . . . . . . 12 (𝑣 ∈ (𝒫 𝐶 ∩ Fin) → 𝑣 ⊆ ({𝐵} ∪ 𝐶))
8685adantl 485 . . . . . . . . . . 11 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑣 ⊆ ({𝐵} ∪ 𝐶))
8780, 86unssd 4116 . . . . . . . . . 10 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → ({𝐵} ∪ 𝑣) ⊆ ({𝐵} ∪ 𝐶))
88 vex 3447 . . . . . . . . . . . 12 𝑣 ∈ V
899, 88unex 7453 . . . . . . . . . . 11 ({𝐵} ∪ 𝑣) ∈ V
9089elpw 4504 . . . . . . . . . 10 (({𝐵} ∪ 𝑣) ∈ 𝒫 ({𝐵} ∪ 𝐶) ↔ ({𝐵} ∪ 𝑣) ⊆ ({𝐵} ∪ 𝐶))
9187, 90sylibr 237 . . . . . . . . 9 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → ({𝐵} ∪ 𝑣) ∈ 𝒫 ({𝐵} ∪ 𝐶))
92 snfi 8581 . . . . . . . . . 10 {𝐵} ∈ Fin
93 inss2 4159 . . . . . . . . . . . 12 (𝒫 𝐶 ∩ Fin) ⊆ Fin
9493sseli 3914 . . . . . . . . . . 11 (𝑣 ∈ (𝒫 𝐶 ∩ Fin) → 𝑣 ∈ Fin)
9594adantl 485 . . . . . . . . . 10 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑣 ∈ Fin)
96 unfi 8773 . . . . . . . . . 10 (({𝐵} ∈ Fin ∧ 𝑣 ∈ Fin) → ({𝐵} ∪ 𝑣) ∈ Fin)
9792, 95, 96sylancr 590 . . . . . . . . 9 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → ({𝐵} ∪ 𝑣) ∈ Fin)
9891, 97elind 4124 . . . . . . . 8 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → ({𝐵} ∪ 𝑣) ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin))
9961eqcomd 2807 . . . . . . . . . . 11 (𝐵𝑉𝐵 = {𝐵})
10099ineq1d 4141 . . . . . . . . . 10 (𝐵𝑉 → (𝐵 𝑣) = ( {𝐵} ∩ 𝑣))
101 intun 4873 . . . . . . . . . 10 ({𝐵} ∪ 𝑣) = ( {𝐵} ∩ 𝑣)
102100, 101eqtr4di 2854 . . . . . . . . 9 (𝐵𝑉 → (𝐵 𝑣) = ({𝐵} ∪ 𝑣))
103102ad3antrrr 729 . . . . . . . 8 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐵 𝑣) = ({𝐵} ∪ 𝑣))
104 inteq 4844 . . . . . . . . 9 (𝑤 = ({𝐵} ∪ 𝑣) → 𝑤 = ({𝐵} ∪ 𝑣))
105104rspceeqv 3589 . . . . . . . 8 ((({𝐵} ∪ 𝑣) ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ (𝐵 𝑣) = ({𝐵} ∪ 𝑣)) → ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)(𝐵 𝑣) = 𝑤)
10698, 103, 105syl2anc 587 . . . . . . 7 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)(𝐵 𝑣) = 𝑤)
107 eqeq1 2805 . . . . . . . 8 (𝐴 = (𝐵 𝑣) → (𝐴 = 𝑤 ↔ (𝐵 𝑣) = 𝑤))
108107rexbidv 3259 . . . . . . 7 (𝐴 = (𝐵 𝑣) → (∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤 ↔ ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)(𝐵 𝑣) = 𝑤))
109106, 108syl5ibrcom 250 . . . . . 6 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐴 = (𝐵 𝑣) → ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤))
110109rexlimdva 3246 . . . . 5 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → (∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣) → ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤))
11178, 110impbid 215 . . . 4 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → (∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤 ↔ ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣)))
11217, 111bitrd 282 . . 3 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) ↔ ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣)))
113112ex 416 . 2 ((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → (𝐴 ∈ V → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) ↔ ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣))))
1142, 7, 113pm5.21ndd 384 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 209  wa 399   = wceq 1538  wcel 2112  wne 2990  wrex 3110  Vcvv 3444  cdif 3881  cun 3882  cin 3883  wss 3884  c0 4246  𝒫 cpw 4500  {csn 4528   cuni 4803   cint 4841  cfv 6328  Fincfn 8496  ficfi 8862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-en 8497  df-fin 8500  df-fi 8863
This theorem is referenced by:  elrfirn  39633
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