Theorem cnvssrndm 5284

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 5140	. . 3 |- Rel `' A
2		relssdmrn 5283	. . 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 4760	. . 3 |- ran A = dom `' A
5		dfdm4 4948	. . 3 |- dom A = ran `' A
6	4, 5	xpeq12i 4771	. 2 |- ( ran A X. dom A ) = ( dom `' A X. ran `' A )
7	3, 6	sseqtrri 3273	1 |- `' A C_ ( ran A X. dom A )

Syntax hints:   C_ wss 3211   X. cxp 4747   `'ccnv 4748   dom cdm 4749   ran crn 4750   Rel wrel 4754

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-xp 4755  df-rel 4756  df-cnv 4757  df-dm 4759  df-rn 4760

This theorem is referenced by: (None)