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Theorem funfvima2d 37990
Description: A function's value in a preimage belongs to the image. (Contributed by Stanislas Polu, 9-Mar-2020.)
Hypothesis
Ref Expression
funfvima2d.1 (𝜑𝐹:𝐴𝐵)
Assertion
Ref Expression
funfvima2d ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ (𝐹𝐴))

Proof of Theorem funfvima2d
StepHypRef Expression
1 funfvima2d.1 . . . 4 (𝜑𝐹:𝐴𝐵)
2 ffun 6015 . . . 4 (𝐹:𝐴𝐵 → Fun 𝐹)
31, 2syl 17 . . 3 (𝜑 → Fun 𝐹)
4 ssid 3609 . . . . 5 𝐴𝐴
54a1i 11 . . . 4 (𝜑𝐴𝐴)
6 fdm 6018 . . . . 5 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
71, 6syl 17 . . . 4 (𝜑 → dom 𝐹 = 𝐴)
85, 7sseqtr4d 3627 . . 3 (𝜑𝐴 ⊆ dom 𝐹)
9 funfvima2 6458 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝑥𝐴 → (𝐹𝑥) ∈ (𝐹𝐴)))
103, 8, 9syl2anc 692 . 2 (𝜑 → (𝑥𝐴 → (𝐹𝑥) ∈ (𝐹𝐴)))
1110imp 445 1 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wss 3560  dom cdm 5084  cima 5087  Fun wfun 5851  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:  imo72b2lem1  37992
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