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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 3484 | . . . . 5 ⊢ 𝑥 ∈ V | |
| 2 | 1 | ideq 5863 | . . . 4 ⊢ (𝑦 I 𝑥 ↔ 𝑦 = 𝑥) |
| 3 | equcom 2017 | . . . 4 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
| 4 | 2, 3 | bitri 275 | . . 3 ⊢ (𝑦 I 𝑥 ↔ 𝑥 = 𝑦) |
| 5 | 4 | opabbii 5210 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑦 I 𝑥} = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} |
| 6 | df-cnv 5693 | . 2 ⊢ ◡ I = {〈𝑥, 𝑦〉 ∣ 𝑦 I 𝑥} | |
| 7 | df-id 5578 | . 2 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | |
| 8 | 5, 6, 7 | 3eqtr4i 2775 | 1 ⊢ ◡ I = I |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 class class class wbr 5143 {copab 5205 I cid 5577 ◡ccnv 5684 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 |
| This theorem is referenced by: coi2 6283 funi 6598 cnvresid 6645 fcoi1 6782 ssdomg 9040 mbfid 25670 mthmpps 35587 brid 38307 extid 38311 cosscnvid 38482 idsymrel 38562 |
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