cnvcnvOLD - Metamath Proof Explorer

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Theorem cnvcnvOLD 5587

Description: Obsolete proof of cnvcnv 5586 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 5503	. . . . 5 ⊢ Rel ^◡^◡𝐴
2		df-rel 5121	. . . . 5 ⊢ (Rel ^◡^◡𝐴 ↔ ^◡^◡𝐴 ⊆ (V × V))
3	1, 2	mpbi 220	. . . 4 ⊢ ^◡^◡𝐴 ⊆ (V × V)
4		relxp 5227	. . . . 5 ⊢ Rel (V × V)
5		dfrel2 5583	. . . . 5 ⊢ (Rel (V × V) ↔ ^◡^◡(V × V) = (V × V))
6	4, 5	mpbi 220	. . . 4 ⊢ ^◡^◡(V × V) = (V × V)
7	3, 6	sseqtr4i 3638	. . 3 ⊢ ^◡^◡𝐴 ⊆ ^◡^◡(V × V)
8		dfss 3589	. . 3 ⊢ (^◡^◡𝐴 ⊆ ^◡^◡(V × V) ↔ ^◡^◡𝐴 = (^◡^◡𝐴 ∩ ^◡^◡(V × V)))
9	7, 8	mpbi 220	. 2 ⊢ ^◡^◡𝐴 = (^◡^◡𝐴 ∩ ^◡^◡(V × V))
10		cnvin 5540	. 2 ⊢ ^◡(^◡𝐴 ∩ ^◡(V × V)) = (^◡^◡𝐴 ∩ ^◡^◡(V × V))
11		cnvin 5540	. . . 4 ⊢ ^◡(𝐴 ∩ (V × V)) = (^◡𝐴 ∩ ^◡(V × V))
12	11	cnveqi 5297	. . 3 ⊢ ^◡^◡(𝐴 ∩ (V × V)) = ^◡(^◡𝐴 ∩ ^◡(V × V))
13		inss2 3834	. . . . 5 ⊢ (𝐴 ∩ (V × V)) ⊆ (V × V)
14		df-rel 5121	. . . . 5 ⊢ (Rel (𝐴 ∩ (V × V)) ↔ (𝐴 ∩ (V × V)) ⊆ (V × V))
15	13, 14	mpbir 221	. . . 4 ⊢ Rel (𝐴 ∩ (V × V))
16		dfrel2 5583	. . . 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 2646	. 2 ⊢ ^◡(^◡𝐴 ∩ ^◡(V × V)) = (𝐴 ∩ (V × V))
19	9, 10, 18	3eqtr2i 2650	1 ⊢ ^◡^◡𝐴 = (𝐴 ∩ (V × V))

Colors of variables: wff setvar class

Syntax hints: = wceq 1483 Vcvv 3200 ∩ cin 3573 ⊆ wss 3574 × cxp 5112 ^◡ccnv 5113 Rel wrel 5119

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122

This theorem is referenced by: (None)

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