Theorem rnoprab2 6744

Description: The range of a restricted operation class abstraction. (Contributed by Scott Fenton, 21-Mar-2012.)

Assertion

Ref	Expression
rnoprab2	|- ran { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) } = { z | E. x e. A E. y e. B ph }

Distinct variable groups:   y, A   x, y, z

Allowed substitution hints:   ph( x, y, z)   A( x, z)   B( x, y, z)

Proof of Theorem rnoprab2

Step	Hyp	Ref	Expression
1		rnoprab 6743	. 2 |- ran { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) } = { z | E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) }
2		r2ex 3061	. . 3 |- ( E. x e. A E. y e. B ph <-> E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) )
3	2	abbii 2739	. 2 |- { z | E. x e. A E. y e. B ph } = { z | E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) }
4	1, 3	eqtr4i 2647	1 |- ran { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) } = { z | E. x e. A E. y e. B ph }

Syntax hints:   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   E.wrex 2913   ran crn 5115   {coprab 6651

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-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-cnv 5122  df-dm 5124  df-rn 5125  df-oprab 6654

This theorem is referenced by:  rnmpt2  6770