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Mirrors > Home > MPE Home > Th. List > fvmptmap | Structured version Visualization version GIF version |
Description: Special case of fvmpt 6754 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 8421 | . 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 6754 | . 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 1537 ∈ wcel 2114 Vcvv 3486 ↦ cmpt 5132 ⟶wf 6337 ‘cfv 6341 (class class class)co 7142 ↑m cmap 8392 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-sbc 3764 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-fv 6349 df-ov 7145 df-oprab 7146 df-mpo 7147 df-map 8394 |
This theorem is referenced by: itg2val 24312 nmopval 29617 nmfnval 29637 eigvecval 29657 eigvalfval 29658 specval 29659 |
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