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Theorem fvelimabd 6720
Description: Deduction form of fvelimab 6719. (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 6719 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶))
41, 2, 3syl2anc 587 1 (𝜑 → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2114  wrex 3131  wss 3908  cima 5535   Fn wfn 6329  cfv 6334
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-fv 6342
This theorem is referenced by:  unima  6721  lmhmima  19810  mdegldg  24665  ig1peu  24770  fnpreimac  30424  fsuppcurry1  30471  fsuppcurry2  30472  swrdrn3  30639  extoimad  40801
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