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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 5701 | . . 3 ⊢ ( I ↾ 𝑋) = ( I ∩ (𝑋 × V)) | |
2 | 1 | eleq2i 2831 | . 2 ⊢ (𝐴 ∈ ( I ↾ 𝑋) ↔ 𝐴 ∈ ( I ∩ (𝑋 × V))) |
3 | elidinxp 6064 | . 2 ⊢ (𝐴 ∈ ( I ∩ (𝑋 × V)) ↔ ∃𝑥 ∈ (𝑋 ∩ V)𝐴 = 〈𝑥, 𝑥〉) | |
4 | inv1 4404 | . . 3 ⊢ (𝑋 ∩ V) = 𝑋 | |
5 | 4 | rexeqi 3323 | . 2 ⊢ (∃𝑥 ∈ (𝑋 ∩ V)𝐴 = 〈𝑥, 𝑥〉 ↔ ∃𝑥 ∈ 𝑋 𝐴 = 〈𝑥, 𝑥〉) |
6 | 2, 3, 5 | 3bitri 297 | 1 ⊢ (𝐴 ∈ ( I ↾ 𝑋) ↔ ∃𝑥 ∈ 𝑋 𝐴 = 〈𝑥, 𝑥〉) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 Vcvv 3478 ∩ cin 3962 〈cop 4637 I cid 5582 × cxp 5687 ↾ cres 5691 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-res 5701 |
This theorem is referenced by: idinxpres 6067 idrefALT 6134 elid 6221 |
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