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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 4206 | . . 3 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐶 | |
2 | imass2 5965 | . . 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 4205 | . . . 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 4171 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴 ∩ 𝐶)) |
10 | fnfvima 6995 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐴 ∩ 𝐶) ⊆ 𝐴 ∧ 𝐵 ∈ (𝐴 ∩ 𝐶)) → (𝐹‘𝐵) ∈ (𝐹 “ (𝐴 ∩ 𝐶))) | |
11 | 4, 6, 9, 10 | syl3anc 1367 | . 2 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (𝐹 “ (𝐴 ∩ 𝐶))) |
12 | 3, 11 | sseldi 3965 | 1 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ∩ cin 3935 ⊆ wss 3936 “ cima 5558 Fn wfn 6350 ‘cfv 6355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-fv 6363 |
This theorem is referenced by: cycpm3cl2 30778 dimkerim 31023 wfximgfd 40534 limsupmnflem 42021 liminfval2 42069 limsup10exlem 42073 liminflelimsupuz 42086 fundcmpsurinjimaid 43591 |
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