Theorem cossxp 5214

Description: Composition as a subset of the cross product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.)

Assertion

Ref	Expression
cossxp	|- ( A o. B ) C_ ( dom B X. ran A )

Proof of Theorem cossxp

Step	Hyp	Ref	Expression
1		relco 5190	. . 3 |- Rel ( A o. B )
2		relssdmrn 5212	. . 3 |- ( Rel ( A o. B ) -> ( A o. B ) C_ ( dom ( A o. B ) X. ran ( A o. B ) ) )
3	1, 2	ax-mp 5	. 2 |- ( A o. B ) C_ ( dom ( A o. B ) X. ran ( A o. B ) )
4		dmcoss 4957	. . 3 |- dom ( A o. B ) C_ dom B
5		rncoss 4958	. . 3 |- ran ( A o. B ) C_ ran A
6		xpss12 4790	. . 3 |- ( ( dom ( A o. B ) C_ dom B /\ ran ( A o. B ) C_ ran A ) -> ( dom ( A o. B ) X. ran ( A o. B ) ) C_ ( dom B X. ran A ) )
7	4, 5, 6	mp2an 426	. 2 |- ( dom ( A o. B ) X. ran ( A o. B ) ) C_ ( dom B X. ran A )
8	3, 7	sstri 3206	1 |- ( A o. B ) C_ ( dom B X. ran A )

Syntax hints:   C_ wss 3170   X. cxp 4681   dom cdm 4683   ran crn 4684   o. ccom 4687   Rel wrel 4688

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-br 4052  df-opab 4114  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694

This theorem is referenced by:  cossxp2  5215  cocnvss  5217  coexg  5236  tposssxp  6348