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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 3451 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | 1 | ideq 5812 | . . . 4 ⊢ (𝑦 I 𝑥 ↔ 𝑦 = 𝑥) |
3 | equcom 2022 | . . . 4 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
4 | 2, 3 | bitri 275 | . . 3 ⊢ (𝑦 I 𝑥 ↔ 𝑥 = 𝑦) |
5 | 4 | opabbii 5176 | . 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑦 I 𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} |
6 | df-cnv 5645 | . 2 ⊢ ◡ I = {⟨𝑥, 𝑦⟩ ∣ 𝑦 I 𝑥} | |
7 | df-id 5535 | . 2 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} | |
8 | 5, 6, 7 | 3eqtr4i 2771 | 1 ⊢ ◡ I = I |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 class class class wbr 5109 {copab 5171 I cid 5534 ◡ccnv 5636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 |
This theorem is referenced by: coi2 6219 funi 6537 cnvresid 6584 fcoi1 6720 ssdomg 8946 mbfid 25022 mthmpps 34240 brid 36817 extid 36821 cosscnvid 36993 idsymrel 37073 |
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