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Theorem funfvima2d 39439
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)
21ffund 6297 . . 3 (𝜑 → Fun 𝐹)
3 ssidd 3843 . . . 4 (𝜑𝐴𝐴)
41fdmd 6302 . . . 4 (𝜑 → dom 𝐹 = 𝐴)
53, 4sseqtr4d 3861 . . 3 (𝜑𝐴 ⊆ dom 𝐹)
6 funfvima2 6767 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝑥𝐴 → (𝐹𝑥) ∈ (𝐹𝐴)))
72, 5, 6syl2anc 579 . 2 (𝜑 → (𝑥𝐴 → (𝐹𝑥) ∈ (𝐹𝐴)))
87imp 397 1 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wcel 2107  wss 3792  dom cdm 5357  cima 5360  Fun wfun 6131  wf 6133  cfv 6137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-fv 6145
This theorem is referenced by:  imo72b2lem1  39441
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