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Theorem elintfv 31788
 Description: Membership in an intersection of function values. (Contributed by Scott Fenton, 9-Dec-2021.)
Hypothesis
Ref Expression
elintfv.1 𝑋 ∈ V
Assertion
Ref Expression
elintfv ((𝐹 Fn 𝐴𝐵𝐴) → (𝑋 (𝐹𝐵) ↔ ∀𝑦𝐵 𝑋 ∈ (𝐹𝑦)))
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑦,𝐹   𝑦,𝑋

Proof of Theorem elintfv
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 elintfv.1 . . 3 𝑋 ∈ V
21elint 4513 . 2 (𝑋 (𝐹𝐵) ↔ ∀𝑧(𝑧 ∈ (𝐹𝐵) → 𝑋𝑧))
3 ssel2 3631 . . . . . . . . . 10 ((𝐵𝐴𝑦𝐵) → 𝑦𝐴)
4 fnbrfvb 6274 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝑦𝐴) → ((𝐹𝑦) = 𝑧𝑦𝐹𝑧))
53, 4sylan2 490 . . . . . . . . 9 ((𝐹 Fn 𝐴 ∧ (𝐵𝐴𝑦𝐵)) → ((𝐹𝑦) = 𝑧𝑦𝐹𝑧))
65anassrs 681 . . . . . . . 8 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝑦𝐵) → ((𝐹𝑦) = 𝑧𝑦𝐹𝑧))
76rexbidva 3078 . . . . . . 7 ((𝐹 Fn 𝐴𝐵𝐴) → (∃𝑦𝐵 (𝐹𝑦) = 𝑧 ↔ ∃𝑦𝐵 𝑦𝐹𝑧))
8 vex 3234 . . . . . . . 8 𝑧 ∈ V
98elima 5506 . . . . . . 7 (𝑧 ∈ (𝐹𝐵) ↔ ∃𝑦𝐵 𝑦𝐹𝑧)
107, 9syl6rbbr 279 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑧 ∈ (𝐹𝐵) ↔ ∃𝑦𝐵 (𝐹𝑦) = 𝑧))
1110imbi1d 330 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝑧 ∈ (𝐹𝐵) → 𝑋𝑧) ↔ (∃𝑦𝐵 (𝐹𝑦) = 𝑧𝑋𝑧)))
12 r19.23v 3052 . . . . 5 (∀𝑦𝐵 ((𝐹𝑦) = 𝑧𝑋𝑧) ↔ (∃𝑦𝐵 (𝐹𝑦) = 𝑧𝑋𝑧))
1311, 12syl6bbr 278 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝑧 ∈ (𝐹𝐵) → 𝑋𝑧) ↔ ∀𝑦𝐵 ((𝐹𝑦) = 𝑧𝑋𝑧)))
1413albidv 1889 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (∀𝑧(𝑧 ∈ (𝐹𝐵) → 𝑋𝑧) ↔ ∀𝑧𝑦𝐵 ((𝐹𝑦) = 𝑧𝑋𝑧)))
15 ralcom4 3255 . . . 4 (∀𝑦𝐵𝑧((𝐹𝑦) = 𝑧𝑋𝑧) ↔ ∀𝑧𝑦𝐵 ((𝐹𝑦) = 𝑧𝑋𝑧))
16 eqcom 2658 . . . . . . . 8 ((𝐹𝑦) = 𝑧𝑧 = (𝐹𝑦))
1716imbi1i 338 . . . . . . 7 (((𝐹𝑦) = 𝑧𝑋𝑧) ↔ (𝑧 = (𝐹𝑦) → 𝑋𝑧))
1817albii 1787 . . . . . 6 (∀𝑧((𝐹𝑦) = 𝑧𝑋𝑧) ↔ ∀𝑧(𝑧 = (𝐹𝑦) → 𝑋𝑧))
19 fvex 6239 . . . . . . 7 (𝐹𝑦) ∈ V
20 eleq2 2719 . . . . . . 7 (𝑧 = (𝐹𝑦) → (𝑋𝑧𝑋 ∈ (𝐹𝑦)))
2119, 20ceqsalv 3264 . . . . . 6 (∀𝑧(𝑧 = (𝐹𝑦) → 𝑋𝑧) ↔ 𝑋 ∈ (𝐹𝑦))
2218, 21bitri 264 . . . . 5 (∀𝑧((𝐹𝑦) = 𝑧𝑋𝑧) ↔ 𝑋 ∈ (𝐹𝑦))
2322ralbii 3009 . . . 4 (∀𝑦𝐵𝑧((𝐹𝑦) = 𝑧𝑋𝑧) ↔ ∀𝑦𝐵 𝑋 ∈ (𝐹𝑦))
2415, 23bitr3i 266 . . 3 (∀𝑧𝑦𝐵 ((𝐹𝑦) = 𝑧𝑋𝑧) ↔ ∀𝑦𝐵 𝑋 ∈ (𝐹𝑦))
2514, 24syl6bb 276 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (∀𝑧(𝑧 ∈ (𝐹𝐵) → 𝑋𝑧) ↔ ∀𝑦𝐵 𝑋 ∈ (𝐹𝑦)))
262, 25syl5bb 272 1 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑋 (𝐹𝐵) ↔ ∀𝑦𝐵 𝑋 ∈ (𝐹𝑦)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383  ∀wal 1521   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942  Vcvv 3231   ⊆ wss 3607  ∩ cint 4507   class class class wbr 4685   “ cima 5146   Fn wfn 5921  ‘cfv 5926 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-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  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-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-id 5053  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-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934 This theorem is referenced by: (None)
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