![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > ovmpod | GIF version |
Description: Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 7-Dec-2014.) |
Ref | Expression |
---|---|
ovmpod.1 | ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)) |
ovmpod.2 | ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆) |
ovmpod.3 | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
ovmpod.4 | ⊢ (𝜑 → 𝐵 ∈ 𝐷) |
ovmpod.5 | ⊢ (𝜑 → 𝑆 ∈ 𝑋) |
Ref | Expression |
---|---|
ovmpod | ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovmpod.1 | . 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)) | |
2 | ovmpod.2 | . 2 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆) | |
3 | eqidd 2178 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐷 = 𝐷) | |
4 | ovmpod.3 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐶) | |
5 | ovmpod.4 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
6 | ovmpod.5 | . 2 ⊢ (𝜑 → 𝑆 ∈ 𝑋) | |
7 | 1, 2, 3, 4, 5, 6 | ovmpodx 5994 | 1 ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 (class class class)co 5868 ∈ cmpo 5870 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-setind 4532 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-iota 5173 df-fun 5213 df-fv 5219 df-ov 5871 df-oprab 5872 df-mpo 5873 |
This theorem is referenced by: ovmpoga 5997 iseqovex 10429 seqvalcd 10432 resqrexlemp1rp 10986 resqrexlemfp1 10989 lcmval 12033 ennnfonelemg 12374 plusfvalg 12661 grpsubval 12796 mulgval 12862 cnfval 13327 cnpfval 13328 blvalps 13521 blval 13522 |
Copyright terms: Public domain | W3C validator |