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Theorem funfvima2d 7229
Description: A function's value in a preimage belongs to the image. (Contributed by Stanislas Polu, 9-Mar-2020.) (Revised by AV, 23-Mar-2024.)
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)
21ffund 6715 . . 3 (𝜑 → Fun 𝐹)
3 ssidd 4000 . . . 4 (𝜑𝐴𝐴)
41fdmd 6722 . . . 4 (𝜑 → dom 𝐹 = 𝐴)
53, 4sseqtrrd 4018 . . 3 (𝜑𝐴 ⊆ dom 𝐹)
6 funfvima2 7228 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝑋𝐴 → (𝐹𝑋) ∈ (𝐹𝐴)))
72, 5, 6syl2anc 583 . 2 (𝜑 → (𝑋𝐴 → (𝐹𝑋) ∈ (𝐹𝐴)))
87imp 406 1 ((𝜑𝑋𝐴) → (𝐹𝑋) ∈ (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2098  wss 3943  dom cdm 5669  cima 5672  Fun wfun 6531  wf 6533  cfv 6537
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545
This theorem is referenced by:  imo72b2lem1  43502  fundcmpsurbijinjpreimafv  46652
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