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Mirrors > Home > ILE Home > Th. List > ovig | GIF version |
Description: The value of an operation class abstraction (weak version). (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 14-Sep-1999.) (Revised by Mario Carneiro, 19-Dec-2013.) |
Ref | Expression |
---|---|
ovig.1 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) |
ovig.2 | ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ∃*𝑧𝜑) |
ovig.3 | ⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)} |
Ref | Expression |
---|---|
ovig | ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐷) → (𝜓 → (𝐴𝐹𝐵) = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3simpa 989 | . 2 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐷) → (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) | |
2 | eleq1 2233 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑅 ↔ 𝐴 ∈ 𝑅)) | |
3 | eleq1 2233 | . . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑆 ↔ 𝐵 ∈ 𝑆)) | |
4 | 2, 3 | bi2anan9 601 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ↔ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆))) |
5 | 4 | 3adant3 1012 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ↔ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆))) |
6 | ovig.1 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) | |
7 | 5, 6 | anbi12d 470 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑) ↔ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ 𝜓))) |
8 | ovig.2 | . . . 4 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ∃*𝑧𝜑) | |
9 | moanimv 2094 | . . . 4 ⊢ (∃*𝑧((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑) ↔ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ∃*𝑧𝜑)) | |
10 | 8, 9 | mpbir 145 | . . 3 ⊢ ∃*𝑧((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑) |
11 | ovig.3 | . . 3 ⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)} | |
12 | 7, 10, 11 | ovigg 5973 | . 2 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐷) → (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ 𝜓) → (𝐴𝐹𝐵) = 𝐶)) |
13 | 1, 12 | mpand 427 | 1 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐷) → (𝜓 → (𝐴𝐹𝐵) = 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 973 = wceq 1348 ∃*wmo 2020 ∈ wcel 2141 (class class class)co 5853 {coprab 5854 |
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-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 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 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-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 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 |
This theorem is referenced by: th3q 6618 addnnnq0 7411 mulnnnq0 7412 addsrpr 7707 mulsrpr 7708 |
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