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Theorem funfvima2d 6968
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 6491 . . 3 (𝜑 → Fun 𝐹)
3 ssidd 3966 . . . 4 (𝜑𝐴𝐴)
41fdmd 6496 . . . 4 (𝜑 → dom 𝐹 = 𝐴)
53, 4sseqtrrd 3984 . . 3 (𝜑𝐴 ⊆ dom 𝐹)
6 funfvima2 6967 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝑋𝐴 → (𝐹𝑋) ∈ (𝐹𝐴)))
72, 5, 6syl2anc 587 . 2 (𝜑 → (𝑋𝐴 → (𝐹𝑋) ∈ (𝐹𝐴)))
87imp 410 1 ((𝜑𝑋𝐴) → (𝐹𝑋) ∈ (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2115  wss 3910  dom cdm 5528  cima 5531  Fun wfun 6322  wf 6324  cfv 6328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336
This theorem is referenced by:  imo72b2lem1  40656  fundcmpsurbijinjpreimafv  43715
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