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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 3478 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | 1 | ideq 5852 | . . . 4 ⊢ (𝑦 I 𝑥 ↔ 𝑦 = 𝑥) |
3 | equcom 2021 | . . . 4 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
4 | 2, 3 | bitri 274 | . . 3 ⊢ (𝑦 I 𝑥 ↔ 𝑥 = 𝑦) |
5 | 4 | opabbii 5215 | . 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑦 I 𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} |
6 | df-cnv 5684 | . 2 ⊢ ◡ I = {⟨𝑥, 𝑦⟩ ∣ 𝑦 I 𝑥} | |
7 | df-id 5574 | . 2 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} | |
8 | 5, 6, 7 | 3eqtr4i 2770 | 1 ⊢ ◡ I = I |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 class class class wbr 5148 {copab 5210 I cid 5573 ◡ccnv 5675 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 |
This theorem is referenced by: coi2 6262 funi 6580 cnvresid 6627 fcoi1 6765 ssdomg 8995 mbfid 25151 mthmpps 34568 brid 37170 extid 37174 cosscnvid 37346 idsymrel 37426 |
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