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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 2797 | . 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆) |
| 4 | ovmpog.3 | . 2 ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) | |
| 5 | 3, 4 | ovmpoga 7587 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ∈ cmpo 7433 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 |
| This theorem is referenced by: ovmpo 7593 naddcllem 8714 mapvalg 8876 pmvalg 8877 genpv 11039 shftfval 15109 efmndov 18894 frlmipval 21799 bcthlem1 25358 negsval 28057 motplusg 28550 signspval 34567 elghomlem1OLD 37892 paddval 39800 tgrpov 40750 erngmul 40808 erngmul-rN 40816 dvamulr 41014 dvavadd 41017 dvhmulr 41088 djavalN 41137 djhval 41400 mendmulr 43196 upfval2 48934 |
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