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Mirrors > Home > MPE Home > Th. List > opelco2g | Structured version Visualization version GIF version |
Description: Ordered pair membership in a composition. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 24-Feb-2015.) |
Ref | Expression |
---|---|
opelco2g | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 ∈ (𝐶 ∘ 𝐷) ↔ ∃𝑥(〈𝐴, 𝑥〉 ∈ 𝐷 ∧ 〈𝑥, 𝐵〉 ∈ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brcog 5720 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))) | |
2 | df-br 5040 | . 2 ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (𝐶 ∘ 𝐷)) | |
3 | df-br 5040 | . . . 4 ⊢ (𝐴𝐷𝑥 ↔ 〈𝐴, 𝑥〉 ∈ 𝐷) | |
4 | df-br 5040 | . . . 4 ⊢ (𝑥𝐶𝐵 ↔ 〈𝑥, 𝐵〉 ∈ 𝐶) | |
5 | 3, 4 | anbi12i 630 | . . 3 ⊢ ((𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵) ↔ (〈𝐴, 𝑥〉 ∈ 𝐷 ∧ 〈𝑥, 𝐵〉 ∈ 𝐶)) |
6 | 5 | exbii 1855 | . 2 ⊢ (∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵) ↔ ∃𝑥(〈𝐴, 𝑥〉 ∈ 𝐷 ∧ 〈𝑥, 𝐵〉 ∈ 𝐶)) |
7 | 1, 2, 6 | 3bitr3g 316 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 ∈ (𝐶 ∘ 𝐷) ↔ ∃𝑥(〈𝐴, 𝑥〉 ∈ 𝐷 ∧ 〈𝑥, 𝐵〉 ∈ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∃wex 1787 ∈ wcel 2112 〈cop 4533 class class class wbr 5039 ∘ ccom 5540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-co 5545 |
This theorem is referenced by: dfco2 6089 dmfco 6785 dfatdmfcoafv2 44361 |
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