Theorem rp-imass 38065

Description: If the R-image of a class A is a subclass of B, then the restriction of R to A is a subset of the Cartesian product of A and B. (Contributed by Richard Penner, 24-Dec-2019.)

Assertion

Ref	Expression
rp-imass	|- ( ( R " A ) C_ B <-> ( R |` A ) C_ ( A X. B ) )

Proof of Theorem rp-imass

Step	Hyp	Ref	Expression
1		df-ima 5127	. . 3 |- ( R " A ) = ran ( R |` A )
2	1	sseq1i 3629	. 2 |- ( ( R " A ) C_ B <-> ran ( R |` A ) C_ B )
3		dmres 5419	. . . 4 |- dom ( R |` A ) = ( A i^i dom R )
4		inss1 3833	. . . 4 |- ( A i^i dom R ) C_ A
5	3, 4	eqsstri 3635	. . 3 |- dom ( R |` A ) C_ A
6	5	biantrur 527	. 2 |- ( ran ( R |` A ) C_ B <-> ( dom ( R |` A ) C_ A /\ ran ( R |` A ) C_ B ) )
7		relres 5426	. . . . 5 |- Rel ( R |` A )
8		relssdmrn 5656	. . . . 5 |- ( Rel ( R |` A ) -> ( R |` A ) C_ ( dom ( R |` A ) X. ran ( R |` A ) ) )
9	7, 8	ax-mp 5	. . . 4 |- ( R |` A ) C_ ( dom ( R |` A ) X. ran ( R |` A ) )
10		xpss12 5225	. . . 4 |- ( ( dom ( R |` A ) C_ A /\ ran ( R |` A ) C_ B ) -> ( dom ( R |` A ) X. ran ( R |` A ) ) C_ ( A X. B ) )
11	9, 10	syl5ss 3614	. . 3 |- ( ( dom ( R |` A ) C_ A /\ ran ( R |` A ) C_ B ) -> ( R |` A ) C_ ( A X. B ) )
12		dmss 5323	. . . . 5 |- ( ( R |` A ) C_ ( A X. B ) -> dom ( R |` A ) C_ dom ( A X. B ) )
13		dmxpss 5565	. . . . 5 |- dom ( A X. B ) C_ A
14	12, 13	syl6ss 3615	. . . 4 |- ( ( R |` A ) C_ ( A X. B ) -> dom ( R |` A ) C_ A )
15		rnss 5354	. . . . 5 |- ( ( R |` A ) C_ ( A X. B ) -> ran ( R |` A ) C_ ran ( A X. B ) )
16		rnxpss 5566	. . . . 5 |- ran ( A X. B ) C_ B
17	15, 16	syl6ss 3615	. . . 4 |- ( ( R |` A ) C_ ( A X. B ) -> ran ( R |` A ) C_ B )
18	14, 17	jca 554	. . 3 |- ( ( R |` A ) C_ ( A X. B ) -> ( dom ( R |` A ) C_ A /\ ran ( R |` A ) C_ B ) )
19	11, 18	impbii 199	. 2 |- ( ( dom ( R |` A ) C_ A /\ ran ( R |` A ) C_ B ) <-> ( R |` A ) C_ ( A X. B ) )
20	2, 6, 19	3bitri 286	1 |- ( ( R " A ) C_ B <-> ( R |` A ) C_ ( A X. B ) )

Syntax hints:   <-> wb 196   /\ wa 384   i^i cin 3573   C_ wss 3574   X. cxp 5112   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   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-ne 2795  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  df-res 5126  df-ima 5127

This theorem is referenced by:  dfhe2  38068