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| Description: Special case of fvmpt 7015 for operator theorems. (Contributed by NM, 27-Nov-2007.) | 
| Ref | Expression | 
|---|---|
| fvmptmap.1 | ⊢ 𝐶 ∈ V | 
| fvmptmap.2 | ⊢ 𝐷 ∈ V | 
| fvmptmap.3 | ⊢ 𝑅 ∈ V | 
| fvmptmap.4 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | 
| fvmptmap.5 | ⊢ 𝐹 = (𝑥 ∈ (𝑅 ↑m 𝐷) ↦ 𝐵) | 
| Ref | Expression | 
|---|---|
| fvmptmap | ⊢ (𝐴:𝐷⟶𝑅 → (𝐹‘𝐴) = 𝐶) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fvmptmap.3 | . . 3 ⊢ 𝑅 ∈ V | |
| 2 | fvmptmap.2 | . . 3 ⊢ 𝐷 ∈ V | |
| 3 | 1, 2 | elmap 8912 | . 2 ⊢ (𝐴 ∈ (𝑅 ↑m 𝐷) ↔ 𝐴:𝐷⟶𝑅) | 
| 4 | fvmptmap.4 | . . 3 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
| 5 | fvmptmap.5 | . . 3 ⊢ 𝐹 = (𝑥 ∈ (𝑅 ↑m 𝐷) ↦ 𝐵) | |
| 6 | fvmptmap.1 | . . 3 ⊢ 𝐶 ∈ V | |
| 7 | 4, 5, 6 | fvmpt 7015 | . 2 ⊢ (𝐴 ∈ (𝑅 ↑m 𝐷) → (𝐹‘𝐴) = 𝐶) | 
| 8 | 3, 7 | sylbir 235 | 1 ⊢ (𝐴:𝐷⟶𝑅 → (𝐹‘𝐴) = 𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ↦ cmpt 5224 ⟶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-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-mpt 5225 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: itg2val 25764 nmopval 31876 nmfnval 31896 eigvecval 31916 eigvalfval 31917 specval 31918 | 
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