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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 5861 | . 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 208 ∧ wa 398 ∈ wcel 2114 Vcvv 3496 〈cop 4575 ↾ cres 5559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-opab 5131 df-xp 5563 df-res 5569 |
This theorem is referenced by: opres 5865 dmres 5877 relssres 5895 iss 5905 restidsing 5924 asymref 5978 ssrnres 6037 cnvresima 6089 ressn 6138 funssres 6400 fcnvres 6558 fvn0ssdmfun 6844 relexpindlem 14424 dprd2dlem1 19165 dprd2da 19166 hausdiag 22255 hauseqlcld 22256 ovoliunlem1 24105 undmrnresiss 39971 |
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