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 8615 | . . . . 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 231 | . . 3 ⊢ (𝜑 → 𝐹:𝐵⟶𝐴) |
9 | 8 | ffvelrnda 6953 | . 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 205 ∈ wcel 2106 ⟶wf 6422 ‘cfv 6426 (class class class)co 7267 ↑m cmap 8602 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3431 df-sbc 3716 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5074 df-opab 5136 df-id 5484 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-fv 6434 df-ov 7270 df-oprab 7271 df-mpo 7272 df-map 8604 |
This theorem is referenced by: ssmapsn 42737 hoidmvle 44119 |
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