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Mirrors > Home > MPE Home > Th. List > fvmptmap | Structured version Visualization version GIF version |
Description: Special case of fvmpt 6762 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 8429 | . 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 6762 | . 2 ⊢ (𝐴 ∈ (𝑅 ↑m 𝐷) → (𝐹‘𝐴) = 𝐶) |
8 | 3, 7 | sylbir 237 | 1 ⊢ (𝐴:𝐷⟶𝑅 → (𝐹‘𝐴) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ↦ cmpt 5138 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ↑m cmap 8400 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-map 8402 |
This theorem is referenced by: itg2val 24323 nmopval 29627 nmfnval 29647 eigvecval 29667 eigvalfval 29668 specval 29669 |
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