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Theorem elrfi 37574
 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 3243 . . 3 (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) → 𝐴 ∈ V)
21a1i 11 . 2 ((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) → 𝐴 ∈ V))
3 inex1g 4834 . . . . 5 (𝐵𝑉 → (𝐵 𝑣) ∈ V)
4 eleq1 2718 . . . . 5 (𝐴 = (𝐵 𝑣) → (𝐴 ∈ V ↔ (𝐵 𝑣) ∈ V))
53, 4syl5ibrcom 237 . . . 4 (𝐵𝑉 → (𝐴 = (𝐵 𝑣) → 𝐴 ∈ V))
65rexlimdvw 3063 . . 3 (𝐵𝑉 → (∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣) → 𝐴 ∈ V))
76adantr 480 . 2 ((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → (∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣) → 𝐴 ∈ V))
8 simpr 476 . . . . 5 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → 𝐴 ∈ V)
9 snex 4938 . . . . . 6 {𝐵} ∈ V
10 pwexg 4880 . . . . . . . 8 (𝐵𝑉 → 𝒫 𝐵 ∈ V)
1110ad2antrr 762 . . . . . . 7 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → 𝒫 𝐵 ∈ V)
12 simplr 807 . . . . . . 7 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → 𝐶 ⊆ 𝒫 𝐵)
1311, 12ssexd 4838 . . . . . 6 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → 𝐶 ∈ V)
14 unexg 7001 . . . . . 6 (({𝐵} ∈ V ∧ 𝐶 ∈ V) → ({𝐵} ∪ 𝐶) ∈ V)
159, 13, 14sylancr 696 . . . . 5 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → ({𝐵} ∪ 𝐶) ∈ V)
16 elfi 8360 . . . . 5 ((𝐴 ∈ V ∧ ({𝐵} ∪ 𝐶) ∈ V) → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) ↔ ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤))
178, 15, 16syl2anc 694 . . . 4 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) ↔ ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤))
18 inss1 3866 . . . . . . . . . . . 12 (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ⊆ 𝒫 ({𝐵} ∪ 𝐶)
19 uncom 3790 . . . . . . . . . . . . 13 ({𝐵} ∪ 𝐶) = (𝐶 ∪ {𝐵})
2019pweqi 4195 . . . . . . . . . . . 12 𝒫 ({𝐵} ∪ 𝐶) = 𝒫 (𝐶 ∪ {𝐵})
2118, 20sseqtri 3670 . . . . . . . . . . 11 (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ⊆ 𝒫 (𝐶 ∪ {𝐵})
2221sseli 3632 . . . . . . . . . 10 (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) → 𝑤 ∈ 𝒫 (𝐶 ∪ {𝐵}))
239elpwun 7019 . . . . . . . . . 10 (𝑤 ∈ 𝒫 (𝐶 ∪ {𝐵}) ↔ (𝑤 ∖ {𝐵}) ∈ 𝒫 𝐶)
2422, 23sylib 208 . . . . . . . . 9 (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) → (𝑤 ∖ {𝐵}) ∈ 𝒫 𝐶)
2524ad2antrl 764 . . . . . . . 8 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → (𝑤 ∖ {𝐵}) ∈ 𝒫 𝐶)
26 inss2 3867 . . . . . . . . . . 11 (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ⊆ Fin
2726sseli 3632 . . . . . . . . . 10 (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) → 𝑤 ∈ Fin)
28 diffi 8233 . . . . . . . . . 10 (𝑤 ∈ Fin → (𝑤 ∖ {𝐵}) ∈ Fin)
2927, 28syl 17 . . . . . . . . 9 (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) → (𝑤 ∖ {𝐵}) ∈ Fin)
3029ad2antrl 764 . . . . . . . 8 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → (𝑤 ∖ {𝐵}) ∈ Fin)
3125, 30elind 3831 . . . . . . 7 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → (𝑤 ∖ {𝐵}) ∈ (𝒫 𝐶 ∩ Fin))
32 incom 3838 . . . . . . . . . . . 12 (𝐵𝐴) = (𝐴𝐵)
33 simprr 811 . . . . . . . . . . . . . 14 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 = 𝑤)
34 simplr 807 . . . . . . . . . . . . . . . . . 18 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 ∈ V)
3533, 34eqeltrrd 2731 . . . . . . . . . . . . . . . . 17 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤 ∈ V)
36 intex 4850 . . . . . . . . . . . . . . . . 17 (𝑤 ≠ ∅ ↔ 𝑤 ∈ V)
3735, 36sylibr 224 . . . . . . . . . . . . . . . 16 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤 ≠ ∅)
38 intssuni 4531 . . . . . . . . . . . . . . . 16 (𝑤 ≠ ∅ → 𝑤 𝑤)
3937, 38syl 17 . . . . . . . . . . . . . . 15 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤 𝑤)
4018sseli 3632 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) → 𝑤 ∈ 𝒫 ({𝐵} ∪ 𝐶))
4140elpwid 4203 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) → 𝑤 ⊆ ({𝐵} ∪ 𝐶))
4241ad2antrl 764 . . . . . . . . . . . . . . . . 17 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤 ⊆ ({𝐵} ∪ 𝐶))
43 pwidg 4206 . . . . . . . . . . . . . . . . . . . . 21 (𝐵𝑉𝐵 ∈ 𝒫 𝐵)
4443snssd 4372 . . . . . . . . . . . . . . . . . . . 20 (𝐵𝑉 → {𝐵} ⊆ 𝒫 𝐵)
4544adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → {𝐵} ⊆ 𝒫 𝐵)
46 simpr 476 . . . . . . . . . . . . . . . . . . 19 ((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → 𝐶 ⊆ 𝒫 𝐵)
4745, 46unssd 3822 . . . . . . . . . . . . . . . . . 18 ((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → ({𝐵} ∪ 𝐶) ⊆ 𝒫 𝐵)
4847ad2antrr 762 . . . . . . . . . . . . . . . . 17 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → ({𝐵} ∪ 𝐶) ⊆ 𝒫 𝐵)
4942, 48sstrd 3646 . . . . . . . . . . . . . . . 16 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤 ⊆ 𝒫 𝐵)
50 sspwuni 4643 . . . . . . . . . . . . . . . 16 (𝑤 ⊆ 𝒫 𝐵 𝑤𝐵)
5149, 50sylib 208 . . . . . . . . . . . . . . 15 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤𝐵)
5239, 51sstrd 3646 . . . . . . . . . . . . . 14 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝑤𝐵)
5333, 52eqsstrd 3672 . . . . . . . . . . . . 13 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴𝐵)
54 df-ss 3621 . . . . . . . . . . . . 13 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
5553, 54sylib 208 . . . . . . . . . . . 12 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → (𝐴𝐵) = 𝐴)
5632, 55syl5req 2698 . . . . . . . . . . 11 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 = (𝐵𝐴))
57 ineq2 3841 . . . . . . . . . . . 12 (𝐴 = 𝑤 → (𝐵𝐴) = (𝐵 𝑤))
5857ad2antll 765 . . . . . . . . . . 11 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → (𝐵𝐴) = (𝐵 𝑤))
5956, 58eqtrd 2685 . . . . . . . . . 10 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 = (𝐵 𝑤))
60 intun 4541 . . . . . . . . . . . 12 ({𝐵} ∪ 𝑤) = ( {𝐵} ∩ 𝑤)
61 intsng 4544 . . . . . . . . . . . . 13 (𝐵𝑉 {𝐵} = 𝐵)
6261ineq1d 3846 . . . . . . . . . . . 12 (𝐵𝑉 → ( {𝐵} ∩ 𝑤) = (𝐵 𝑤))
6360, 62syl5req 2698 . . . . . . . . . . 11 (𝐵𝑉 → (𝐵 𝑤) = ({𝐵} ∪ 𝑤))
6463ad3antrrr 766 . . . . . . . . . 10 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → (𝐵 𝑤) = ({𝐵} ∪ 𝑤))
6559, 64eqtrd 2685 . . . . . . . . 9 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 = ({𝐵} ∪ 𝑤))
66 undif2 4077 . . . . . . . . . 10 ({𝐵} ∪ (𝑤 ∖ {𝐵})) = ({𝐵} ∪ 𝑤)
6766inteqi 4511 . . . . . . . . 9 ({𝐵} ∪ (𝑤 ∖ {𝐵})) = ({𝐵} ∪ 𝑤)
6865, 67syl6eqr 2703 . . . . . . . 8 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 = ({𝐵} ∪ (𝑤 ∖ {𝐵})))
69 intun 4541 . . . . . . . . . 10 ({𝐵} ∪ (𝑤 ∖ {𝐵})) = ( {𝐵} ∩ (𝑤 ∖ {𝐵}))
7061ineq1d 3846 . . . . . . . . . 10 (𝐵𝑉 → ( {𝐵} ∩ (𝑤 ∖ {𝐵})) = (𝐵 (𝑤 ∖ {𝐵})))
7169, 70syl5eq 2697 . . . . . . . . 9 (𝐵𝑉 ({𝐵} ∪ (𝑤 ∖ {𝐵})) = (𝐵 (𝑤 ∖ {𝐵})))
7271ad3antrrr 766 . . . . . . . 8 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → ({𝐵} ∪ (𝑤 ∖ {𝐵})) = (𝐵 (𝑤 ∖ {𝐵})))
7368, 72eqtrd 2685 . . . . . . 7 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → 𝐴 = (𝐵 (𝑤 ∖ {𝐵})))
74 inteq 4510 . . . . . . . . . 10 (𝑣 = (𝑤 ∖ {𝐵}) → 𝑣 = (𝑤 ∖ {𝐵}))
7574ineq2d 3847 . . . . . . . . 9 (𝑣 = (𝑤 ∖ {𝐵}) → (𝐵 𝑣) = (𝐵 (𝑤 ∖ {𝐵})))
7675eqeq2d 2661 . . . . . . . 8 (𝑣 = (𝑤 ∖ {𝐵}) → (𝐴 = (𝐵 𝑣) ↔ 𝐴 = (𝐵 (𝑤 ∖ {𝐵}))))
7776rspcev 3340 . . . . . . 7 (((𝑤 ∖ {𝐵}) ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝐴 = (𝐵 (𝑤 ∖ {𝐵}))) → ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣))
7831, 73, 77syl2anc 694 . . . . . 6 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ 𝐴 = 𝑤)) → ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣))
7978rexlimdvaa 3061 . . . . 5 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → (∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤 → ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣)))
80 ssun1 3809 . . . . . . . . . . . 12 {𝐵} ⊆ ({𝐵} ∪ 𝐶)
8180a1i 11 . . . . . . . . . . 11 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → {𝐵} ⊆ ({𝐵} ∪ 𝐶))
82 inss1 3866 . . . . . . . . . . . . . 14 (𝒫 𝐶 ∩ Fin) ⊆ 𝒫 𝐶
8382sseli 3632 . . . . . . . . . . . . 13 (𝑣 ∈ (𝒫 𝐶 ∩ Fin) → 𝑣 ∈ 𝒫 𝐶)
84 elpwi 4201 . . . . . . . . . . . . 13 (𝑣 ∈ 𝒫 𝐶𝑣𝐶)
85 ssun4 3812 . . . . . . . . . . . . 13 (𝑣𝐶𝑣 ⊆ ({𝐵} ∪ 𝐶))
8683, 84, 853syl 18 . . . . . . . . . . . 12 (𝑣 ∈ (𝒫 𝐶 ∩ Fin) → 𝑣 ⊆ ({𝐵} ∪ 𝐶))
8786adantl 481 . . . . . . . . . . 11 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑣 ⊆ ({𝐵} ∪ 𝐶))
8881, 87unssd 3822 . . . . . . . . . 10 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → ({𝐵} ∪ 𝑣) ⊆ ({𝐵} ∪ 𝐶))
89 vex 3234 . . . . . . . . . . . 12 𝑣 ∈ V
909, 89unex 6998 . . . . . . . . . . 11 ({𝐵} ∪ 𝑣) ∈ V
9190elpw 4197 . . . . . . . . . 10 (({𝐵} ∪ 𝑣) ∈ 𝒫 ({𝐵} ∪ 𝐶) ↔ ({𝐵} ∪ 𝑣) ⊆ ({𝐵} ∪ 𝐶))
9288, 91sylibr 224 . . . . . . . . 9 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → ({𝐵} ∪ 𝑣) ∈ 𝒫 ({𝐵} ∪ 𝐶))
93 snfi 8079 . . . . . . . . . 10 {𝐵} ∈ Fin
94 inss2 3867 . . . . . . . . . . . 12 (𝒫 𝐶 ∩ Fin) ⊆ Fin
9594sseli 3632 . . . . . . . . . . 11 (𝑣 ∈ (𝒫 𝐶 ∩ Fin) → 𝑣 ∈ Fin)
9695adantl 481 . . . . . . . . . 10 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → 𝑣 ∈ Fin)
97 unfi 8268 . . . . . . . . . 10 (({𝐵} ∈ Fin ∧ 𝑣 ∈ Fin) → ({𝐵} ∪ 𝑣) ∈ Fin)
9893, 96, 97sylancr 696 . . . . . . . . 9 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → ({𝐵} ∪ 𝑣) ∈ Fin)
9992, 98elind 3831 . . . . . . . 8 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → ({𝐵} ∪ 𝑣) ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin))
10061eqcomd 2657 . . . . . . . . . . 11 (𝐵𝑉𝐵 = {𝐵})
101100ineq1d 3846 . . . . . . . . . 10 (𝐵𝑉 → (𝐵 𝑣) = ( {𝐵} ∩ 𝑣))
102 intun 4541 . . . . . . . . . 10 ({𝐵} ∪ 𝑣) = ( {𝐵} ∩ 𝑣)
103101, 102syl6eqr 2703 . . . . . . . . 9 (𝐵𝑉 → (𝐵 𝑣) = ({𝐵} ∪ 𝑣))
104103ad3antrrr 766 . . . . . . . 8 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐵 𝑣) = ({𝐵} ∪ 𝑣))
105 inteq 4510 . . . . . . . . . 10 (𝑤 = ({𝐵} ∪ 𝑣) → 𝑤 = ({𝐵} ∪ 𝑣))
106105eqeq2d 2661 . . . . . . . . 9 (𝑤 = ({𝐵} ∪ 𝑣) → ((𝐵 𝑣) = 𝑤 ↔ (𝐵 𝑣) = ({𝐵} ∪ 𝑣)))
107106rspcev 3340 . . . . . . . 8 ((({𝐵} ∪ 𝑣) ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin) ∧ (𝐵 𝑣) = ({𝐵} ∪ 𝑣)) → ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)(𝐵 𝑣) = 𝑤)
10899, 104, 107syl2anc 694 . . . . . . 7 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)(𝐵 𝑣) = 𝑤)
109 eqeq1 2655 . . . . . . . 8 (𝐴 = (𝐵 𝑣) → (𝐴 = 𝑤 ↔ (𝐵 𝑣) = 𝑤))
110109rexbidv 3081 . . . . . . 7 (𝐴 = (𝐵 𝑣) → (∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤 ↔ ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)(𝐵 𝑣) = 𝑤))
111108, 110syl5ibrcom 237 . . . . . 6 ((((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) ∧ 𝑣 ∈ (𝒫 𝐶 ∩ Fin)) → (𝐴 = (𝐵 𝑣) → ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤))
112111rexlimdva 3060 . . . . 5 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → (∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣) → ∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤))
11379, 112impbid 202 . . . 4 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → (∃𝑤 ∈ (𝒫 ({𝐵} ∪ 𝐶) ∩ Fin)𝐴 = 𝑤 ↔ ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣)))
11417, 113bitrd 268 . . 3 (((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) ∧ 𝐴 ∈ V) → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) ↔ ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣)))
115114ex 449 . 2 ((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → (𝐴 ∈ V → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) ↔ ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣))))
1162, 7, 115pm5.21ndd 368 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 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∃wrex 2942  Vcvv 3231   ∖ cdif 3604   ∪ cun 3605   ∩ cin 3606   ⊆ wss 3607  ∅c0 3948  𝒫 cpw 4191  {csn 4210  ∪ cuni 4468  ∩ cint 4507  ‘cfv 5926  Fincfn 7997  ficfi 8357 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-fin 8001  df-fi 8358 This theorem is referenced by:  elrfirn  37575
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