Theorem intimass 37946

Description: The image under the intersection of relations is a subset of the intersection of the images. (Contributed by RP, 13-Apr-2020.)

Assertion

Ref	Expression
intimass	|- ( |^| A " B ) C_ |^| { x | E. a e. A x = ( a " B ) }

Distinct variable groups:   x, a, A   B, a, x

Proof of Theorem intimass

Dummy variables y b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.12 3063	. . 3 |- ( E. b e. B A. a e. A <. b , y >. e. a -> A. a e. A E. b e. B <. b , y >. e. a )
2		elimaint 37940	. . 3 |- ( y e. ( |^| A " B ) <-> E. b e. B A. a e. A <. b , y >. e. a )
3		elintima 37945	. . 3 |- ( y e. |^| { x | E. a e. A x = ( a " B ) } <-> A. a e. A E. b e. B <. b , y >. e. a )
4	1, 2, 3	3imtr4i 281	. 2 |- ( y e. ( |^| A " B ) -> y e. |^| { x | E. a e. A x = ( a " B ) } )
5	4	ssriv 3607	1 |- ( |^| A " B ) C_ |^| { x | E. a e. A x = ( a " B ) }

Syntax hints:   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   C_ wss 3574   <.cop 4183   |^|cint 4475   "cima 5117

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-8 1992  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  ax-un 6949

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-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  intimass2  37947