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Theorem wfximgfd 37984
Description: The value of a function on its domain is in the image of the function. (Contributed by Stanislas Polu, 9-Mar-2020.)
Hypotheses
Ref Expression
wfximgfd.1 (𝜑𝐶𝐴)
wfximgfd.2 (𝜑𝐹:𝐴𝐵)
Assertion
Ref Expression
wfximgfd (𝜑 → (𝐹𝐶) ∈ (𝐹𝐴))

Proof of Theorem wfximgfd
StepHypRef Expression
1 wfximgfd.2 . . 3 (𝜑𝐹:𝐴𝐵)
2 ffn 6012 . . 3 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
31, 2syl 17 . 2 (𝜑𝐹 Fn 𝐴)
4 ssid 3609 . . 3 𝐴𝐴
54a1i 11 . 2 (𝜑𝐴𝐴)
6 wfximgfd.1 . 2 (𝜑𝐶𝐴)
7 fnfvima 6461 . 2 ((𝐹 Fn 𝐴𝐴𝐴𝐶𝐴) → (𝐹𝐶) ∈ (𝐹𝐴))
83, 5, 6, 7syl3anc 1323 1 (𝜑 → (𝐹𝐶) ∈ (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  wss 3560  cima 5087   Fn wfn 5852  wf 5853  cfv 5857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865
This theorem is referenced by:  imo72b2lem0  37986
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