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Theorem funfvima2d 37491
 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)
2 ffun 5961 . . . 4 (𝐹:𝐴𝐵 → Fun 𝐹)
31, 2syl 17 . . 3 (𝜑 → Fun 𝐹)
4 ssid 3587 . . . . 5 𝐴𝐴
54a1i 11 . . . 4 (𝜑𝐴𝐴)
6 fdm 5964 . . . . 5 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
71, 6syl 17 . . . 4 (𝜑 → dom 𝐹 = 𝐴)
85, 7sseqtr4d 3605 . . 3 (𝜑𝐴 ⊆ dom 𝐹)
9 funfvima2 6397 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝑥𝐴 → (𝐹𝑥) ∈ (𝐹𝐴)))
103, 8, 9syl2anc 691 . 2 (𝜑 → (𝑥𝐴 → (𝐹𝑥) ∈ (𝐹𝐴)))
1110imp 444 1 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ (𝐹𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  dom cdm 5038   “ cima 5041  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812 This theorem is referenced by:  imo72b2lem1  37493
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