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Theorem fvelimabd 6967
Description: Deduction form of fvelimab 6966. (Contributed by Stanislas Polu, 9-Mar-2020.)
Hypotheses
Ref Expression
fvelimabd.1 (𝜑𝐹 Fn 𝐴)
fvelimabd.2 (𝜑𝐵𝐴)
Assertion
Ref Expression
fvelimabd (𝜑 → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐶   𝑥,𝐹
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem fvelimabd
StepHypRef Expression
1 fvelimabd.1 . 2 (𝜑𝐹 Fn 𝐴)
2 fvelimabd.2 . 2 (𝜑𝐵𝐴)
3 fvelimab 6966 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶))
41, 2, 3syl2anc 582 1 (𝜑 → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wcel 2098  wrex 3060  wss 3939  cima 5675   Fn wfn 6538  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551
This theorem is referenced by:  unima  6968  ghmquskerlem1  19238  lmhmima  20936  mdegldg  26020  ig1peu  26127  2ndimaxp  32478  fnpreimac  32502  fsuppcurry1  32552  fsuppcurry2  32553  swrdrn3  32721  ghmqusnsglem1  33178  fnrelpredd  34769  bj-gabima  36475  extoimad  43659
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