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Mathbox for Stanislas Polu |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wfximgfd | Structured version Visualization version GIF version |
Description: The value of a function on its domain is in the image of the function. (Contributed by Stanislas Polu, 9-Mar-2020.) |
Ref | Expression |
---|---|
wfximgfd.1 | ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
wfximgfd.2 | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
Ref | Expression |
---|---|
wfximgfd | ⊢ (𝜑 → (𝐹‘𝐶) ∈ (𝐹 “ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wfximgfd.2 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
2 | ffn 5958 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
4 | ssid 3587 | . . 3 ⊢ 𝐴 ⊆ 𝐴 | |
5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐴) |
6 | wfximgfd.1 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝐴) | |
7 | fnfvima 6400 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ (𝐹 “ 𝐴)) | |
8 | 3, 5, 6, 7 | syl3anc 1318 | 1 ⊢ (𝜑 → (𝐹‘𝐶) ∈ (𝐹 “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ⊆ wss 3540 “ cima 5041 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 |
This theorem is referenced by: imo72b2lem0 37487 |
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