Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fvmap | Structured version Visualization version GIF version |
Description: Function value for a member of a set exponentiation. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
Ref | Expression |
---|---|
fvmap.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
fvmap.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
fvmap.f | ⊢ (𝜑 → 𝐹 ∈ (𝐴 ↑𝑚 𝐵)) |
fvmap.c | ⊢ (𝜑 → 𝐶 ∈ 𝐵) |
Ref | Expression |
---|---|
fvmap | ⊢ (𝜑 → (𝐹‘𝐶) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 ⊢ (𝜑 → 𝜑) | |
2 | fvmap.c | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝐵) | |
3 | fvmap.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐴 ↑𝑚 𝐵)) | |
4 | fvmap.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
5 | fvmap.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
6 | elmapg 7757 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹 ∈ (𝐴 ↑𝑚 𝐵) ↔ 𝐹:𝐵⟶𝐴)) | |
7 | 4, 5, 6 | syl2anc 691 | . . . 4 ⊢ (𝜑 → (𝐹 ∈ (𝐴 ↑𝑚 𝐵) ↔ 𝐹:𝐵⟶𝐴)) |
8 | 3, 7 | mpbid 221 | . . 3 ⊢ (𝜑 → 𝐹:𝐵⟶𝐴) |
9 | 8 | ffvelrnda 6267 | . 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐵) → (𝐹‘𝐶) ∈ 𝐴) |
10 | 1, 2, 9 | syl2anc 691 | 1 ⊢ (𝜑 → (𝐹‘𝐶) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∈ wcel 1977 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 |
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-8 1979 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-pow 4769 ax-pr 4833 ax-un 6847 |
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-pw 4110 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-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 |
This theorem is referenced by: ssmapsn 38403 hoidmvle 39490 |
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