Theorem cnvssrndm 6125

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 ⊆ (ran 𝐴 × dom 𝐴)

Proof of Theorem cnvssrndm

Step	Hyp	Ref	Expression
1		relcnv 5970	. . 3 ⊢ Rel ◡𝐴
2		relssdmrn 6124	. . 3 ⊢ (Rel ◡𝐴 → ◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴))
3	1, 2	ax-mp 5	. 2 ⊢ ◡𝐴 ⊆ (dom ◡𝐴 × ran ◡𝐴)
4		df-rn 5569	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
5		dfdm4 5767	. . 3 ⊢ dom 𝐴 = ran ◡𝐴
6	4, 5	xpeq12i 5586	. 2 ⊢ (ran 𝐴 × dom 𝐴) = (dom ◡𝐴 × ran ◡𝐴)
7	3, 6	sseqtrri 4007	1 ⊢ ◡𝐴 ⊆ (ran 𝐴 × dom 𝐴)

Syntax hints:   ⊆ wss 3939   × cxp 5556  ^◡ccnv 5557  dom cdm 5558  ran crn 5559  Rel wrel 5563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-br 5070  df-opab 5132  df-xp 5564  df-rel 5565  df-cnv 5566  df-dm 5568  df-rn 5569

This theorem is referenced by:  wuncnv  10155  fcnvgreu  30421  trclubgNEW  39984