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Mirrors > Home > MPE Home > Th. List > ovmpog | 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 |
---|---|
ovmpog.1 | ⊢ (𝑥 = 𝐴 → 𝑅 = 𝐺) |
ovmpog.2 | ⊢ (𝑦 = 𝐵 → 𝐺 = 𝑆) |
ovmpog.3 | ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) |
Ref | Expression |
---|---|
ovmpog | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovmpog.1 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑅 = 𝐺) | |
2 | ovmpog.2 | . . 3 ⊢ (𝑦 = 𝐵 → 𝐺 = 𝑆) | |
3 | 1, 2 | sylan9eq 2798 | . 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆) |
4 | ovmpog.3 | . 2 ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) | |
5 | 3, 4 | ovmpoga 7427 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 (class class class)co 7275 ∈ cmpo 7277 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-iota 6391 df-fun 6435 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 |
This theorem is referenced by: ovmpo 7433 mapvalg 8625 pmvalg 8626 genpv 10755 shftfval 14781 efmndov 18520 frlmipval 20986 bcthlem1 24488 motplusg 26903 signspval 32531 naddcllem 33831 negsval 34123 elghomlem1OLD 36043 paddval 37812 tgrpov 38762 erngmul 38820 erngmul-rN 38828 dvamulr 39026 dvavadd 39029 dvhmulr 39100 djavalN 39149 djhval 39412 mendmulr 41013 |
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