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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 6740 . . 3 (𝜑 → Fun 𝐹)
3 ssidd 4007 . . . 4 (𝜑𝐴𝐴)
41fdmd 6746 . . . 4 (𝜑 → dom 𝐹 = 𝐴)
53, 4sseqtrrd 4021 . . 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 2108  wss 3951  dom cdm 5685  cima 5688  Fun wfun 6555  wf 6557  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569
This theorem is referenced by:  imo72b2lem1  44182  fundcmpsurbijinjpreimafv  47394
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