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Theorem funfvima2d 7252
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 6741 . . 3 (𝜑 → Fun 𝐹)
3 ssidd 4019 . . . 4 (𝜑𝐴𝐴)
41fdmd 6747 . . . 4 (𝜑 → dom 𝐹 = 𝐴)
53, 4sseqtrrd 4037 . . 3 (𝜑𝐴 ⊆ dom 𝐹)
6 funfvima2 7251 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝑋𝐴 → (𝐹𝑋) ∈ (𝐹𝐴)))
72, 5, 6syl2anc 584 . 2 (𝜑 → (𝑋𝐴 → (𝐹𝑋) ∈ (𝐹𝐴)))
87imp 406 1 ((𝜑𝑋𝐴) → (𝐹𝑋) ∈ (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  wss 3963  dom cdm 5689  cima 5692  Fun wfun 6557  wf 6559  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571
This theorem is referenced by:  imo72b2lem1  44159  fundcmpsurbijinjpreimafv  47332
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