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Mirrors > Home > MPE Home > Th. List > oprabex3 | Structured version Visualization version 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 7456 | . 2 ⊢ (𝐻 × 𝐻) ∈ V |
3 | moeq 3646 | . . . . . 6 ⊢ ∃*𝑧 𝑧 = 𝑅 | |
4 | 3 | mosubop 5366 | . . . . 5 ⊢ ∃*𝑧∃𝑢∃𝑓(𝑦 = 〈𝑢, 𝑓〉 ∧ 𝑧 = 𝑅) |
5 | 4 | mosubop 5366 | . . . 4 ⊢ ∃*𝑧∃𝑤∃𝑣(𝑥 = 〈𝑤, 𝑣〉 ∧ ∃𝑢∃𝑓(𝑦 = 〈𝑢, 𝑓〉 ∧ 𝑧 = 𝑅)) |
6 | anass 472 | . . . . . . . 8 ⊢ (((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 𝑅) ↔ (𝑥 = 〈𝑤, 𝑣〉 ∧ (𝑦 = 〈𝑢, 𝑓〉 ∧ 𝑧 = 𝑅))) | |
7 | 6 | 2exbii 1850 | . . . . . . 7 ⊢ (∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 𝑅) ↔ ∃𝑢∃𝑓(𝑥 = 〈𝑤, 𝑣〉 ∧ (𝑦 = 〈𝑢, 𝑓〉 ∧ 𝑧 = 𝑅))) |
8 | 19.42vv 1958 | . . . . . . 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 278 | . . . . . 6 ⊢ (∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 𝑅) ↔ (𝑥 = 〈𝑤, 𝑣〉 ∧ ∃𝑢∃𝑓(𝑦 = 〈𝑢, 𝑓〉 ∧ 𝑧 = 𝑅))) |
10 | 9 | 2exbii 1850 | . . . . 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 2606 | . . . 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 234 | . . 3 ⊢ ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 𝑅) |
13 | 12 | a1i 11 | . 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 7659 | 1 ⊢ 𝐹 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ∃*wmo 2596 Vcvv 3441 〈cop 4531 × cxp 5517 {coprab 7136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-oprab 7139 |
This theorem is referenced by: (None) |
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