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Theorem funfvima2d 38971
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 6209 . . . 4 (𝐹:𝐴𝐵 → Fun 𝐹)
31, 2syl 17 . . 3 (𝜑 → Fun 𝐹)
4 ssid 3765 . . . . 5 𝐴𝐴
54a1i 11 . . . 4 (𝜑𝐴𝐴)
6 fdm 6212 . . . . 5 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
71, 6syl 17 . . . 4 (𝜑 → dom 𝐹 = 𝐴)
85, 7sseqtr4d 3783 . . 3 (𝜑𝐴 ⊆ dom 𝐹)
9 funfvima2 6656 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝑥𝐴 → (𝐹𝑥) ∈ (𝐹𝐴)))
103, 8, 9syl2anc 696 . 2 (𝜑 → (𝑥𝐴 → (𝐹𝑥) ∈ (𝐹𝐴)))
1110imp 444 1 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wss 3715  dom cdm 5266  cima 5269  Fun wfun 6043  wf 6045  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057
This theorem is referenced by:  imo72b2lem1  38973
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