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Theorem fnfvimad 6986
Description: A function's value belongs to the image. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
fnfvimad.1 (𝜑𝐹 Fn 𝐴)
fnfvimad.2 (𝜑𝐵𝐴)
fnfvimad.3 (𝜑𝐵𝐶)
Assertion
Ref Expression
fnfvimad (𝜑 → (𝐹𝐵) ∈ (𝐹𝐶))

Proof of Theorem fnfvimad
StepHypRef Expression
1 inss2 4191 . . 3 (𝐴𝐶) ⊆ 𝐶
2 imass2 5953 . . 3 ((𝐴𝐶) ⊆ 𝐶 → (𝐹 “ (𝐴𝐶)) ⊆ (𝐹𝐶))
31, 2ax-mp 5 . 2 (𝐹 “ (𝐴𝐶)) ⊆ (𝐹𝐶)
4 fnfvimad.1 . . 3 (𝜑𝐹 Fn 𝐴)
5 inss1 4190 . . . 4 (𝐴𝐶) ⊆ 𝐴
65a1i 11 . . 3 (𝜑 → (𝐴𝐶) ⊆ 𝐴)
7 fnfvimad.2 . . . 4 (𝜑𝐵𝐴)
8 fnfvimad.3 . . . 4 (𝜑𝐵𝐶)
97, 8elind 4156 . . 3 (𝜑𝐵 ∈ (𝐴𝐶))
10 fnfvima 6985 . . 3 ((𝐹 Fn 𝐴 ∧ (𝐴𝐶) ⊆ 𝐴𝐵 ∈ (𝐴𝐶)) → (𝐹𝐵) ∈ (𝐹 “ (𝐴𝐶)))
114, 6, 9, 10syl3anc 1368 . 2 (𝜑 → (𝐹𝐵) ∈ (𝐹 “ (𝐴𝐶)))
123, 11sseldi 3951 1 (𝜑 → (𝐹𝐵) ∈ (𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2115  cin 3918  wss 3919  cima 5546   Fn wfn 6339  cfv 6344
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 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318
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 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-fv 6352
This theorem is referenced by:  cycpm3cl2  30805  dimkerim  31053  wfximgfd  40726  limsupmnflem  42228  liminfval2  42276  limsup10exlem  42280  liminflelimsupuz  42293  fundcmpsurinjimaid  43794
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