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Mirrors > Home > MPE Home > Th. List > cnvi | Structured version Visualization version GIF version |
Description: The converse of the identity relation. Theorem 3.7(ii) of [Monk1] p. 36. (Contributed by NM, 26-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
cnvi | ⊢ ◡ I = I |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3482 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | 1 | ideq 5866 | . . . 4 ⊢ (𝑦 I 𝑥 ↔ 𝑦 = 𝑥) |
3 | equcom 2015 | . . . 4 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
4 | 2, 3 | bitri 275 | . . 3 ⊢ (𝑦 I 𝑥 ↔ 𝑥 = 𝑦) |
5 | 4 | opabbii 5215 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑦 I 𝑥} = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} |
6 | df-cnv 5697 | . 2 ⊢ ◡ I = {〈𝑥, 𝑦〉 ∣ 𝑦 I 𝑥} | |
7 | df-id 5583 | . 2 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | |
8 | 5, 6, 7 | 3eqtr4i 2773 | 1 ⊢ ◡ I = I |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 class class class wbr 5148 {copab 5210 I cid 5582 ◡ccnv 5688 |
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-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-cnv 5697 |
This theorem is referenced by: coi2 6285 funi 6600 cnvresid 6647 fcoi1 6783 ssdomg 9039 mbfid 25684 mthmpps 35567 brid 38288 extid 38292 cosscnvid 38463 idsymrel 38543 |
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