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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 5824 | . 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 209 ∧ wa 399 ∈ wcel 2111 Vcvv 3441 〈cop 4531 ↾ cres 5521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-opab 5093 df-xp 5525 df-res 5531 |
This theorem is referenced by: opres 5828 dmres 5840 relssres 5859 iss 5870 restidsing 5889 asymref 5943 ssrnres 6002 cnvresima 6054 ressn 6104 funssres 6368 fcnvres 6530 fvn0ssdmfun 6819 relexpindlem 14414 dprd2dlem1 19156 dprd2da 19157 hausdiag 22250 hauseqlcld 22251 ovoliunlem1 24106 undmrnresiss 40304 |
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