![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fvmptmap | Structured version Visualization version GIF version |
Description: Special case of fvmpt 6529 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 8151 | . 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 6529 | . 2 ⊢ (𝐴 ∈ (𝑅 ↑𝑚 𝐷) → (𝐹‘𝐴) = 𝐶) |
8 | 3, 7 | sylbir 227 | 1 ⊢ (𝐴:𝐷⟶𝑅 → (𝐹‘𝐴) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 Vcvv 3414 ↦ cmpt 4952 ⟶wf 6119 ‘cfv 6123 (class class class)co 6905 ↑𝑚 cmap 8122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-map 8124 |
This theorem is referenced by: itg2val 23894 nmopval 29270 nmfnval 29290 eigvecval 29310 eigvalfval 29311 specval 29312 |
Copyright terms: Public domain | W3C validator |