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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 7773 | . 2 ⊢ (𝐻 × 𝐻) ∈ V | 
| 3 | moeq 3713 | . . . . . 6 ⊢ ∃*𝑧 𝑧 = 𝑅 | |
| 4 | 3 | mosubop 5516 | . . . . 5 ⊢ ∃*𝑧∃𝑢∃𝑓(𝑦 = 〈𝑢, 𝑓〉 ∧ 𝑧 = 𝑅) | 
| 5 | 4 | mosubop 5516 | . . . 4 ⊢ ∃*𝑧∃𝑤∃𝑣(𝑥 = 〈𝑤, 𝑣〉 ∧ ∃𝑢∃𝑓(𝑦 = 〈𝑢, 𝑓〉 ∧ 𝑧 = 𝑅)) | 
| 6 | anass 468 | . . . . . . . 8 ⊢ (((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 𝑅) ↔ (𝑥 = 〈𝑤, 𝑣〉 ∧ (𝑦 = 〈𝑢, 𝑓〉 ∧ 𝑧 = 𝑅))) | |
| 7 | 6 | 2exbii 1849 | . . . . . . 7 ⊢ (∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 𝑅) ↔ ∃𝑢∃𝑓(𝑥 = 〈𝑤, 𝑣〉 ∧ (𝑦 = 〈𝑢, 𝑓〉 ∧ 𝑧 = 𝑅))) | 
| 8 | 19.42vv 1957 | . . . . . . 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 275 | . . . . . 6 ⊢ (∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 𝑅) ↔ (𝑥 = 〈𝑤, 𝑣〉 ∧ ∃𝑢∃𝑓(𝑦 = 〈𝑢, 𝑓〉 ∧ 𝑧 = 𝑅))) | 
| 10 | 9 | 2exbii 1849 | . . . . 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 2548 | . . . 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 231 | . . 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 8001 | 1 ⊢ 𝐹 ∈ V | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∃*wmo 2538 Vcvv 3480 〈cop 4632 × cxp 5683 {coprab 7432 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-oprab 7435 | 
| This theorem is referenced by: (None) | 
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