Theorem ssoprab2i 6034

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 342	. . . 4 |- ( ( w = <. x , y >. /\ ph ) -> ( w = <. x , y >. /\ ps ) )
3	2	2eximi 1624	. . 3 |- ( E. x E. y ( w = <. x , y >. /\ ph ) -> E. x E. y ( w = <. x , y >. /\ ps ) )
4	3	ssopab2i 4324	. 2 |- { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) } C_ { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ps ) }
5		dfoprab2 5992	. 2 |- { <. <. x , y >. , z >. | ph } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }
6		dfoprab2 5992	. 2 |- { <. <. x , y >. , z >. | ps } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ps ) }
7	4, 5, 6	3sstr4i 3234	1 |- { <. <. x , y >. , z >. | ph } C_ { <. <. x , y >. , z >. | ps }

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   E.wex 1515   C_ wss 3166   <.cop 3636   {copab 4104   {coprab 5945

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-opab 4106  df-oprab 5948

This theorem is referenced by:  mpomulf  8062