Theorem ssoprab2i 5931

Description: Inference of operation class abstraction subclass from implication. (Contributed by NM, 11-Nov-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Hypothesis

Ref	Expression
ssoprab2i.1	|- ( ph -> ps )

Assertion

Ref	Expression
ssoprab2i	|- { <. <. x , y >. , z >. | ph } C_ { <. <. x , y >. , z >. | ps }

Distinct variable groups:   x, z   y, z

Allowed substitution hints:   ph( x, y, z)   ps( x, y, z)

Proof of Theorem ssoprab2i

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssoprab2i.1	. . . . 5 |- ( ph -> ps )
2	1	anim2i 340	. . . 4 |- ( ( w = <. x , y >. /\ ph ) -> ( w = <. x , y >. /\ ps ) )
3	2	2eximi 1589	. . 3 |- ( E. x E. y ( w = <. x , y >. /\ ph ) -> E. x E. y ( w = <. x , y >. /\ ps ) )
4	3	ssopab2i 4255	. 2 |- { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) } C_ { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ps ) }
5		dfoprab2 5889	. 2 |- { <. <. x , y >. , z >. | ph } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }
6		dfoprab2 5889	. 2 |- { <. <. x , y >. , z >. | ps } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ps ) }
7	4, 5, 6	3sstr4i 3183	1 |- { <. <. x , y >. , z >. | ph } C_ { <. <. x , y >. , z >. | ps }

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   E.wex 1480   C_ wss 3116   <.cop 3579   {copab 4042   {coprab 5843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-opab 4044  df-oprab 5846

This theorem is referenced by: (None)