cnvcnvOLD - Metamath Proof Explorer

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Theorem cnvcnvOLD 5622

Description: Obsolete proof of cnvcnv 5621 as of 26-Nov-2021. (Contributed by NM, 8-Dec-2003.) (Proof modification is discouraged.) (New usage is discouraged.)

**Assertion**
Ref	Expression
cnvcnvOLD	⊢ ^◡^◡𝐴 = (𝐴 ∩ (V × V))

**Proof of Theorem cnvcnvOLD**
Step	Hyp	Ref	Expression
1		relcnv 5538	. . . . 5 ⊢ Rel ^◡^◡𝐴
2		df-rel 5150	. . . . 5 ⊢ (Rel ^◡^◡𝐴 ↔ ^◡^◡𝐴 ⊆ (V × V))
3	1, 2	mpbi 220	. . . 4 ⊢ ^◡^◡𝐴 ⊆ (V × V)
4		relxp 5160	. . . . 5 ⊢ Rel (V × V)
5		dfrel2 5618	. . . . 5 ⊢ (Rel (V × V) ↔ ^◡^◡(V × V) = (V × V))
6	4, 5	mpbi 220	. . . 4 ⊢ ^◡^◡(V × V) = (V × V)
7	3, 6	sseqtr4i 3671	. . 3 ⊢ ^◡^◡𝐴 ⊆ ^◡^◡(V × V)
8		dfss 3622	. . 3 ⊢ (^◡^◡𝐴 ⊆ ^◡^◡(V × V) ↔ ^◡^◡𝐴 = (^◡^◡𝐴 ∩ ^◡^◡(V × V)))
9	7, 8	mpbi 220	. 2 ⊢ ^◡^◡𝐴 = (^◡^◡𝐴 ∩ ^◡^◡(V × V))
10		cnvin 5575	. 2 ⊢ ^◡(^◡𝐴 ∩ ^◡(V × V)) = (^◡^◡𝐴 ∩ ^◡^◡(V × V))
11		cnvin 5575	. . . 4 ⊢ ^◡(𝐴 ∩ (V × V)) = (^◡𝐴 ∩ ^◡(V × V))
12	11	cnveqi 5329	. . 3 ⊢ ^◡^◡(𝐴 ∩ (V × V)) = ^◡(^◡𝐴 ∩ ^◡(V × V))
13		inss2 3867	. . . . 5 ⊢ (𝐴 ∩ (V × V)) ⊆ (V × V)
14		df-rel 5150	. . . . 5 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) ⊆ (V × V))
15	13, 14	mpbir 221	. . . 4 ⊢ Rel (𝐴 ∩ (V × V))
16		dfrel2 5618	. . . 4 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ ^◡^◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V)))
17	15, 16	mpbi 220	. . 3 ⊢ ^◡^◡(𝐴 ∩ (V × V)) = (𝐴 ∩ (V × V))
18	12, 17	eqtr3i 2675	. 2 ⊢ ^◡(^◡𝐴 ∩ ^◡(V × V)) = (𝐴 ∩ (V × V))
19	9, 10, 18	3eqtr2i 2679	1 ⊢ ^◡^◡𝐴 = (𝐴 ∩ (V × V))

Colors of variables: wff setvar class

Syntax hints: = wceq 1523 Vcvv 3231 ∩ cin 3606 ⊆ wss 3607 × cxp 5141 ^◡ccnv 5142 Rel wrel 5148

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-br 4686 df-opab 4746 df-xp 5149 df-rel 5150 df-cnv 5151

This theorem is referenced by: (None)

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