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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 8878 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹 ∈ (𝐴 ↑m 𝐵) ↔ 𝐹:𝐵⟶𝐴)) | |
7 | 4, 5, 6 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝐹 ∈ (𝐴 ↑m 𝐵) ↔ 𝐹:𝐵⟶𝐴)) |
8 | 3, 7 | mpbid 232 | . . 3 ⊢ (𝜑 → 𝐹:𝐵⟶𝐴) |
9 | 8 | ffvelcdmda 7104 | . 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐵) → (𝐹‘𝐶) ∈ 𝐴) |
10 | 1, 2, 9 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐹‘𝐶) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2106 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8867 |
This theorem is referenced by: ssmapsn 45159 hoidmvle 46556 |
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