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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 2171 | . 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 5979 | 1 ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∈ wcel 2141 (class class class)co 5853 ∈ cmpo 5855 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 |
This theorem is referenced by: ovmpoga 5982 iseqovex 10412 seqvalcd 10415 resqrexlemp1rp 10970 resqrexlemfp1 10973 lcmval 12017 ennnfonelemg 12358 plusfvalg 12617 grpsubval 12749 cnfval 12988 cnpfval 12989 blvalps 13182 blval 13183 |
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