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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 2664 | . 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆) |
4 | ovmpt2g.3 | . 2 ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) | |
5 | 3, 4 | ovmpt2ga 6688 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 (class class class)co 6549 ↦ cmpt2 6551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 |
This theorem is referenced by: ovmpt2 6694 mapvalg 7754 pmvalg 7755 cdaval 8875 genpv 9700 shftfval 13658 symgov 17633 frlmipval 19937 bcthlem1 22929 motplusg 25237 signspval 29955 elghomlem1OLD 32854 paddval 34102 tgrpov 35054 erngmul 35112 erngmul-rN 35120 dvamulr 35318 dvavadd 35321 dvhmulr 35393 djavalN 35442 djhval 35705 mendmulr 36777 |
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