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| Mirrors > Home > MPE Home > Th. List > opelresi | Structured version Visualization version GIF version | ||
| Description: Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.) |
| Ref | Expression |
|---|---|
| opelresi.1 | ⊢ 𝐶 ∈ V |
| Ref | Expression |
|---|---|
| opelresi | ⊢ (〈𝐵, 𝐶〉 ∈ (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 〈𝐵, 𝐶〉 ∈ 𝑅)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | opelresi.1 | . 2 ⊢ 𝐶 ∈ V | |
| 2 | opelres 6003 | . 2 ⊢ (𝐶 ∈ V → (〈𝐵, 𝐶〉 ∈ (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 〈𝐵, 𝐶〉 ∈ 𝑅))) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ (〈𝐵, 𝐶〉 ∈ (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 〈𝐵, 𝐶〉 ∈ 𝑅)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2108 Vcvv 3480 〈cop 4632 ↾ cres 5687 |
| 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-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-opab 5206 df-xp 5691 df-res 5697 |
| This theorem is referenced by: opres 6007 dmres 6030 relssres 6040 iss 6053 restidsing 6071 asymref 6136 ssrnres 6198 cnvresima 6250 ressn 6305 funssres 6610 fcnvres 6785 fvn0ssdmfun 7094 relexpindlem 15102 dprd2dlem1 20061 dprd2da 20062 hausdiag 23653 hauseqlcld 23654 ovoliunlem1 25537 undmrnresiss 43617 |
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