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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 6860 | . 2 ⊢ (𝐻 × 𝐻) ∈ V |
3 | moeq 3349 | . . . . . 6 ⊢ ∃*𝑧 𝑧 = 𝑅 | |
4 | 3 | mosubop 4898 | . . . . 5 ⊢ ∃*𝑧∃𝑢∃𝑓(𝑦 = 〈𝑢, 𝑓〉 ∧ 𝑧 = 𝑅) |
5 | 4 | mosubop 4898 | . . . 4 ⊢ ∃*𝑧∃𝑤∃𝑣(𝑥 = 〈𝑤, 𝑣〉 ∧ ∃𝑢∃𝑓(𝑦 = 〈𝑢, 𝑓〉 ∧ 𝑧 = 𝑅)) |
6 | anass 679 | . . . . . . . 8 ⊢ (((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 𝑅) ↔ (𝑥 = 〈𝑤, 𝑣〉 ∧ (𝑦 = 〈𝑢, 𝑓〉 ∧ 𝑧 = 𝑅))) | |
7 | 6 | 2exbii 1765 | . . . . . . 7 ⊢ (∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 𝑅) ↔ ∃𝑢∃𝑓(𝑥 = 〈𝑤, 𝑣〉 ∧ (𝑦 = 〈𝑢, 𝑓〉 ∧ 𝑧 = 𝑅))) |
8 | 19.42vv 1907 | . . . . . . 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 263 | . . . . . 6 ⊢ (∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 𝑅) ↔ (𝑥 = 〈𝑤, 𝑣〉 ∧ ∃𝑢∃𝑓(𝑦 = 〈𝑢, 𝑓〉 ∧ 𝑧 = 𝑅))) |
10 | 9 | 2exbii 1765 | . . . . 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 2481 | . . . 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 220 | . . 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 7047 | 1 ⊢ 𝐹 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∃*wmo 2459 Vcvv 3173 〈cop 4131 × cxp 5036 {coprab 6550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-oprab 6553 |
This theorem is referenced by: (None) |
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