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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 4190 | . . 3 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐶 | |
2 | imass2 6055 | . . 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 4189 | . . . 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 4155 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴 ∩ 𝐶)) |
10 | fnfvima 7184 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐴 ∩ 𝐶) ⊆ 𝐴 ∧ 𝐵 ∈ (𝐴 ∩ 𝐶)) → (𝐹‘𝐵) ∈ (𝐹 “ (𝐴 ∩ 𝐶))) | |
11 | 4, 6, 9, 10 | syl3anc 1372 | . 2 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (𝐹 “ (𝐴 ∩ 𝐶))) |
12 | 3, 11 | sselid 3943 | 1 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ∩ cin 3910 ⊆ wss 3911 “ cima 5637 Fn wfn 6492 ‘cfv 6497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-fv 6505 |
This theorem is referenced by: cycpm3cl2 32034 rhmimaidl 32254 dimkerim 32379 wfximgfd 42524 limsupmnflem 44047 liminfval2 44095 limsup10exlem 44099 liminflelimsupuz 44112 fundcmpsurinjimaid 45689 |
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