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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 8880 | . . . . 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 7103 | . 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 2107 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ↑m cmap 8867 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-map 8869 | 
| This theorem is referenced by: ssmapsn 45226 hoidmvle 46620 | 
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