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Mirrors > Home > MPE Home > Th. List > elrid | Structured version Visualization version GIF version |
Description: Characterization of the elements of a restricted identity relation. (Contributed by BJ, 28-Aug-2022.) (Proof shortened by Peter Mazsa, 9-Sep-2022.) |
Ref | Expression |
---|---|
elrid | ⊢ (𝐴 ∈ ( I ↾ 𝑋) ↔ ∃𝑥 ∈ 𝑋 𝐴 = 〈𝑥, 𝑥〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-res 5531 | . . 3 ⊢ ( I ↾ 𝑋) = ( I ∩ (𝑋 × V)) | |
2 | 1 | eleq2i 2881 | . 2 ⊢ (𝐴 ∈ ( I ↾ 𝑋) ↔ 𝐴 ∈ ( I ∩ (𝑋 × V))) |
3 | elidinxp 5878 | . 2 ⊢ (𝐴 ∈ ( I ∩ (𝑋 × V)) ↔ ∃𝑥 ∈ (𝑋 ∩ V)𝐴 = 〈𝑥, 𝑥〉) | |
4 | inv1 4302 | . . 3 ⊢ (𝑋 ∩ V) = 𝑋 | |
5 | 4 | rexeqi 3363 | . 2 ⊢ (∃𝑥 ∈ (𝑋 ∩ V)𝐴 = 〈𝑥, 𝑥〉 ↔ ∃𝑥 ∈ 𝑋 𝐴 = 〈𝑥, 𝑥〉) |
6 | 2, 3, 5 | 3bitri 300 | 1 ⊢ (𝐴 ∈ ( I ↾ 𝑋) ↔ ∃𝑥 ∈ 𝑋 𝐴 = 〈𝑥, 𝑥〉) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 Vcvv 3441 ∩ cin 3880 〈cop 4531 I cid 5424 × cxp 5517 ↾ 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-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 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-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-res 5531 |
This theorem is referenced by: idinxpres 5881 idrefALT 5940 elid 6023 |
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