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Mirrors > Home > ILE Home > Th. List > oprabex3 | GIF version |
Description: Existence of an operation class abstraction (special case). (Contributed by NM, 19-Oct-2004.) |
Ref | Expression |
---|---|
oprabex3.1 | ⊢ 𝐻 ∈ V |
oprabex3.2 | ⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 𝑅))} |
Ref | Expression |
---|---|
oprabex3 | ⊢ 𝐹 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oprabex3.1 | . . 3 ⊢ 𝐻 ∈ V | |
2 | 1, 1 | xpex 4724 | . 2 ⊢ (𝐻 × 𝐻) ∈ V |
3 | moeq 2905 | . . . . . 6 ⊢ ∃*𝑧 𝑧 = 𝑅 | |
4 | 3 | mosubop 4675 | . . . . 5 ⊢ ∃*𝑧∃𝑢∃𝑓(𝑦 = 〈𝑢, 𝑓〉 ∧ 𝑧 = 𝑅) |
5 | 4 | mosubop 4675 | . . . 4 ⊢ ∃*𝑧∃𝑤∃𝑣(𝑥 = 〈𝑤, 𝑣〉 ∧ ∃𝑢∃𝑓(𝑦 = 〈𝑢, 𝑓〉 ∧ 𝑧 = 𝑅)) |
6 | anass 399 | . . . . . . . 8 ⊢ (((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 𝑅) ↔ (𝑥 = 〈𝑤, 𝑣〉 ∧ (𝑦 = 〈𝑢, 𝑓〉 ∧ 𝑧 = 𝑅))) | |
7 | 6 | 2exbii 1599 | . . . . . . 7 ⊢ (∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 𝑅) ↔ ∃𝑢∃𝑓(𝑥 = 〈𝑤, 𝑣〉 ∧ (𝑦 = 〈𝑢, 𝑓〉 ∧ 𝑧 = 𝑅))) |
8 | 19.42vv 1904 | . . . . . . 7 ⊢ (∃𝑢∃𝑓(𝑥 = 〈𝑤, 𝑣〉 ∧ (𝑦 = 〈𝑢, 𝑓〉 ∧ 𝑧 = 𝑅)) ↔ (𝑥 = 〈𝑤, 𝑣〉 ∧ ∃𝑢∃𝑓(𝑦 = 〈𝑢, 𝑓〉 ∧ 𝑧 = 𝑅))) | |
9 | 7, 8 | bitri 183 | . . . . . 6 ⊢ (∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 𝑅) ↔ (𝑥 = 〈𝑤, 𝑣〉 ∧ ∃𝑢∃𝑓(𝑦 = 〈𝑢, 𝑓〉 ∧ 𝑧 = 𝑅))) |
10 | 9 | 2exbii 1599 | . . . . 5 ⊢ (∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 𝑅) ↔ ∃𝑤∃𝑣(𝑥 = 〈𝑤, 𝑣〉 ∧ ∃𝑢∃𝑓(𝑦 = 〈𝑢, 𝑓〉 ∧ 𝑧 = 𝑅))) |
11 | 10 | mobii 2056 | . . . 4 ⊢ (∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 𝑅) ↔ ∃*𝑧∃𝑤∃𝑣(𝑥 = 〈𝑤, 𝑣〉 ∧ ∃𝑢∃𝑓(𝑦 = 〈𝑢, 𝑓〉 ∧ 𝑧 = 𝑅))) |
12 | 5, 11 | mpbir 145 | . . 3 ⊢ ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 𝑅) |
13 | 12 | a1i 9 | . 2 ⊢ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) → ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 𝑅)) |
14 | oprabex3.2 | . 2 ⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 𝑅))} | |
15 | 2, 2, 13, 14 | oprabex 6105 | 1 ⊢ 𝐹 ∈ V |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1348 ∃wex 1485 ∃*wmo 2020 ∈ wcel 2141 Vcvv 2730 〈cop 3584 × cxp 4607 {coprab 5852 |
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-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 |
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-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-oprab 5855 |
This theorem is referenced by: addvalex 7795 |
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