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Mirrors > Home > MPE Home > Th. List > fnfvimad | Structured version Visualization version GIF version |
Description: A function's value belongs to the image. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
fnfvimad.1 | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
fnfvimad.2 | ⊢ (𝜑 → 𝐵 ∈ 𝐴) |
fnfvimad.3 | ⊢ (𝜑 → 𝐵 ∈ 𝐶) |
Ref | Expression |
---|---|
fnfvimad | ⊢ (𝜑 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss2 4188 | . . 3 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐶 | |
2 | imass2 6053 | . . 3 ⊢ ((𝐴 ∩ 𝐶) ⊆ 𝐶 → (𝐹 “ (𝐴 ∩ 𝐶)) ⊆ (𝐹 “ 𝐶)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (𝐹 “ (𝐴 ∩ 𝐶)) ⊆ (𝐹 “ 𝐶) |
4 | fnfvimad.1 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
5 | inss1 4187 | . . . 4 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐴 | |
6 | 5 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐶) ⊆ 𝐴) |
7 | fnfvimad.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐴) | |
8 | fnfvimad.3 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐶) | |
9 | 7, 8 | elind 4153 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴 ∩ 𝐶)) |
10 | fnfvima 7180 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐴 ∩ 𝐶) ⊆ 𝐴 ∧ 𝐵 ∈ (𝐴 ∩ 𝐶)) → (𝐹‘𝐵) ∈ (𝐹 “ (𝐴 ∩ 𝐶))) | |
11 | 4, 6, 9, 10 | syl3anc 1371 | . 2 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (𝐹 “ (𝐴 ∩ 𝐶))) |
12 | 3, 11 | sselid 3941 | 1 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐶)) |
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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