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Theorem elintfv 34378
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 4918 . 2 (𝑋 (𝐹𝐵) ↔ ∀𝑧(𝑧 ∈ (𝐹𝐵) → 𝑋𝑧))
3 fvelimab 6919 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑧 ∈ (𝐹𝐵) ↔ ∃𝑦𝐵 (𝐹𝑦) = 𝑧))
43imbi1d 342 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝑧 ∈ (𝐹𝐵) → 𝑋𝑧) ↔ (∃𝑦𝐵 (𝐹𝑦) = 𝑧𝑋𝑧)))
5 r19.23v 3180 . . . . 5 (∀𝑦𝐵 ((𝐹𝑦) = 𝑧𝑋𝑧) ↔ (∃𝑦𝐵 (𝐹𝑦) = 𝑧𝑋𝑧))
64, 5bitr4di 289 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝑧 ∈ (𝐹𝐵) → 𝑋𝑧) ↔ ∀𝑦𝐵 ((𝐹𝑦) = 𝑧𝑋𝑧)))
76albidv 1924 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (∀𝑧(𝑧 ∈ (𝐹𝐵) → 𝑋𝑧) ↔ ∀𝑧𝑦𝐵 ((𝐹𝑦) = 𝑧𝑋𝑧)))
8 ralcom4 3272 . . . 4 (∀𝑦𝐵𝑧((𝐹𝑦) = 𝑧𝑋𝑧) ↔ ∀𝑧𝑦𝐵 ((𝐹𝑦) = 𝑧𝑋𝑧))
9 eqcom 2744 . . . . . . . 8 ((𝐹𝑦) = 𝑧𝑧 = (𝐹𝑦))
109imbi1i 350 . . . . . . 7 (((𝐹𝑦) = 𝑧𝑋𝑧) ↔ (𝑧 = (𝐹𝑦) → 𝑋𝑧))
1110albii 1822 . . . . . 6 (∀𝑧((𝐹𝑦) = 𝑧𝑋𝑧) ↔ ∀𝑧(𝑧 = (𝐹𝑦) → 𝑋𝑧))
12 fvex 6860 . . . . . . 7 (𝐹𝑦) ∈ V
13 eleq2 2827 . . . . . . 7 (𝑧 = (𝐹𝑦) → (𝑋𝑧𝑋 ∈ (𝐹𝑦)))
1412, 13ceqsalv 3484 . . . . . 6 (∀𝑧(𝑧 = (𝐹𝑦) → 𝑋𝑧) ↔ 𝑋 ∈ (𝐹𝑦))
1511, 14bitri 275 . . . . 5 (∀𝑧((𝐹𝑦) = 𝑧𝑋𝑧) ↔ 𝑋 ∈ (𝐹𝑦))
1615ralbii 3097 . . . 4 (∀𝑦𝐵𝑧((𝐹𝑦) = 𝑧𝑋𝑧) ↔ ∀𝑦𝐵 𝑋 ∈ (𝐹𝑦))
178, 16bitr3i 277 . . 3 (∀𝑧𝑦𝐵 ((𝐹𝑦) = 𝑧𝑋𝑧) ↔ ∀𝑦𝐵 𝑋 ∈ (𝐹𝑦))
187, 17bitrdi 287 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (∀𝑧(𝑧 ∈ (𝐹𝐵) → 𝑋𝑧) ↔ ∀𝑦𝐵 𝑋 ∈ (𝐹𝑦)))
192, 18bitrid 283 1 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑋 (𝐹𝐵) ↔ ∀𝑦𝐵 𝑋 ∈ (𝐹𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540   = wceq 1542  wcel 2107  wral 3065  wrex 3074  Vcvv 3448  wss 3915   cint 4912  cima 5641   Fn wfn 6496  cfv 6501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-fv 6509
This theorem is referenced by: (None)
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