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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 5731 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))) | |
2 | df-br 5059 | . 2 ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (𝐶 ∘ 𝐷)) | |
3 | df-br 5059 | . . . 4 ⊢ (𝐴𝐷𝑥 ↔ 〈𝐴, 𝑥〉 ∈ 𝐷) | |
4 | df-br 5059 | . . . 4 ⊢ (𝑥𝐶𝐵 ↔ 〈𝑥, 𝐵〉 ∈ 𝐶) | |
5 | 3, 4 | anbi12i 628 | . . 3 ⊢ ((𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵) ↔ (〈𝐴, 𝑥〉 ∈ 𝐷 ∧ 〈𝑥, 𝐵〉 ∈ 𝐶)) |
6 | 5 | exbii 1844 | . 2 ⊢ (∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵) ↔ ∃𝑥(〈𝐴, 𝑥〉 ∈ 𝐷 ∧ 〈𝑥, 𝐵〉 ∈ 𝐶)) |
7 | 1, 2, 6 | 3bitr3g 315 | 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 208 ∧ wa 398 ∃wex 1776 ∈ wcel 2110 〈cop 4566 class class class wbr 5058 ∘ ccom 5553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-co 5558 |
This theorem is referenced by: dfco2 6092 dmfco 6751 dfatdmfcoafv2 43447 |
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