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Mirrors > Home > MPE Home > Th. List > ovmpt2g | Structured version Visualization version GIF version |
Description: Value of an operation given by a maps-to rule. Special case. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.) |
Ref | Expression |
---|---|
ovmpt2g.1 | ⊢ (𝑥 = 𝐴 → 𝑅 = 𝐺) |
ovmpt2g.2 | ⊢ (𝑦 = 𝐵 → 𝐺 = 𝑆) |
ovmpt2g.3 | ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) |
Ref | Expression |
---|---|
ovmpt2g | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovmpt2g.1 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑅 = 𝐺) | |
2 | ovmpt2g.2 | . . 3 ⊢ (𝑦 = 𝐵 → 𝐺 = 𝑆) | |
3 | 1, 2 | sylan9eq 2814 | . 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆) |
4 | ovmpt2g.3 | . 2 ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) | |
5 | 3, 4 | ovmpt2ga 6955 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 (class class class)co 6813 ↦ cmpt2 6815 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-iota 6012 df-fun 6051 df-fv 6057 df-ov 6816 df-oprab 6817 df-mpt2 6818 |
This theorem is referenced by: ovmpt2 6961 mapvalg 8033 pmvalg 8034 cdaval 9184 genpv 10013 shftfval 14009 symgov 18010 frlmipval 20320 bcthlem1 23321 motplusg 25636 signspval 30938 elghomlem1OLD 33997 paddval 35587 tgrpov 36538 erngmul 36596 erngmul-rN 36604 dvamulr 36802 dvavadd 36805 dvhmulr 36877 djavalN 36926 djhval 37189 mendmulr 38260 |
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