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Theorem fnfvimad 7181
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 4188 . . 3 (𝐴𝐶) ⊆ 𝐶
2 imass2 6053 . . 3 ((𝐴𝐶) ⊆ 𝐶 → (𝐹 “ (𝐴𝐶)) ⊆ (𝐹𝐶))
31, 2ax-mp 5 . 2 (𝐹 “ (𝐴𝐶)) ⊆ (𝐹𝐶)
4 fnfvimad.1 . . 3 (𝜑𝐹 Fn 𝐴)
5 inss1 4187 . . . 4 (𝐴𝐶) ⊆ 𝐴
65a1i 11 . . 3 (𝜑 → (𝐴𝐶) ⊆ 𝐴)
7 fnfvimad.2 . . . 4 (𝜑𝐵𝐴)
8 fnfvimad.3 . . . 4 (𝜑𝐵𝐶)
97, 8elind 4153 . . 3 (𝜑𝐵 ∈ (𝐴𝐶))
10 fnfvima 7180 . . 3 ((𝐹 Fn 𝐴 ∧ (𝐴𝐶) ⊆ 𝐴𝐵 ∈ (𝐴𝐶)) → (𝐹𝐵) ∈ (𝐹 “ (𝐴𝐶)))
114, 6, 9, 10syl3anc 1371 . 2 (𝜑 → (𝐹𝐵) ∈ (𝐹 “ (𝐴𝐶)))
123, 11sselid 3941 1 (𝜑 → (𝐹𝐵) ∈ (𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  cin 3908  wss 3909  cima 5635   Fn wfn 6489  cfv 6494
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2707  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6446  df-fun 6496  df-fn 6497  df-fv 6502
This theorem is referenced by:  cycpm3cl2  31880  rhmimaidl  32097  dimkerim  32213  wfximgfd  42416  limsupmnflem  43931  liminfval2  43979  limsup10exlem  43983  liminflelimsupuz  43996  fundcmpsurinjimaid  45573
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