Theorem cnvssrndm 5657

Description: The converse is a subset of the cartesian product of range and domain. (Contributed by Mario Carneiro, 2-Jan-2017.)

Assertion

Ref	Expression
cnvssrndm	|- `' A C_ ( ran A X. dom A )

Proof of Theorem cnvssrndm

Step	Hyp	Ref	Expression
1		relcnv 5503	. . 3 |- Rel `' A
2		relssdmrn 5656	. . 3 |- ( Rel `' A -> `' A C_ ( dom `' A X. ran `' A ) )
3	1, 2	ax-mp 5	. 2 |- `' A C_ ( dom `' A X. ran `' A )
4		df-rn 5125	. . 3 |- ran A = dom `' A
5		dfdm4 5316	. . 3 |- dom A = ran `' A
6	4, 5	xpeq12i 5137	. 2 |- ( ran A X. dom A ) = ( dom `' A X. ran `' A )
7	3, 6	sseqtr4i 3638	1 |- `' A C_ ( ran A X. dom A )

Syntax hints:   C_ wss 3574   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   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-rex 2918  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  df-dm 5124  df-rn 5125

This theorem is referenced by:  wuncnv  9552  fcnvgreu  29472  trclubgNEW  37925