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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 2793 | . 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆) |
4 | ovmpog.3 | . 2 ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) | |
5 | 3, 4 | ovmpoga 7319 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 (class class class)co 7170 ∈ cmpo 7172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-iota 6297 df-fun 6341 df-fv 6347 df-ov 7173 df-oprab 7174 df-mpo 7175 |
This theorem is referenced by: ovmpo 7325 mapvalg 8447 pmvalg 8448 genpv 10499 shftfval 14519 efmndov 18162 frlmipval 20595 bcthlem1 24076 motplusg 26488 signspval 32101 naddcllem 33474 negsval 33766 elghomlem1OLD 35686 paddval 37455 tgrpov 38405 erngmul 38463 erngmul-rN 38471 dvamulr 38669 dvavadd 38672 dvhmulr 38743 djavalN 38792 djhval 39055 mendmulr 40605 |
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