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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 3474 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | 1 | ideq 5850 | . . . 4 ⊢ (𝑦 I 𝑥 ↔ 𝑦 = 𝑥) |
3 | equcom 2014 | . . . 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 5681 | . 2 ⊢ ◡ I = {⟨𝑥, 𝑦⟩ ∣ 𝑦 I 𝑥} | |
7 | df-id 5571 | . 2 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} | |
8 | 5, 6, 7 | 3eqtr4i 2766 | 1 ⊢ ◡ I = I |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 class class class wbr 5143 {copab 5205 I cid 5570 ◡ccnv 5672 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-br 5144 df-opab 5206 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 |
This theorem is referenced by: coi2 6262 funi 6580 cnvresid 6627 fcoi1 6766 ssdomg 9015 mbfid 25558 mthmpps 35187 brid 37773 extid 37777 cosscnvid 37948 idsymrel 38028 |
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