Theorem ex-opab 27289

Description: Example for df-opab 4713. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

Assertion

Ref	Expression
ex-opab	|- ( R = { <. x , y >. | ( x e. CC /\ y e. CC /\ ( x + 1 ) = y ) } -> 3 R 4 )

Distinct variable group:   x, y

Allowed substitution hints:   R( x, y)

Proof of Theorem ex-opab

Step	Hyp	Ref	Expression
1		3cn 11095	. . 3 |- 3 e. CC
2		4cn 11098	. . 3 |- 4 e. CC
3		3p1e4 11153	. . 3 |- ( 3 + 1 ) = 4
4	1	elexi 3213	. . . 4 |- 3 e. _V
5	2	elexi 3213	. . . 4 |- 4 e. _V
6		eleq1 2689	. . . . 5 |- ( x = 3 -> ( x e. CC <-> 3 e. CC ) )
7		oveq1 6657	. . . . . 6 |- ( x = 3 -> ( x + 1 ) = ( 3 + 1 ) )
8	7	eqeq1d 2624	. . . . 5 |- ( x = 3 -> ( ( x + 1 ) = y <-> ( 3 + 1 ) = y ) )
9	6, 8	3anbi13d 1401	. . . 4 |- ( x = 3 -> ( ( x e. CC /\ y e. CC /\ ( x + 1 ) = y ) <-> ( 3 e. CC /\ y e. CC /\ ( 3 + 1 ) = y ) ) )
10		eleq1 2689	. . . . 5 |- ( y = 4 -> ( y e. CC <-> 4 e. CC ) )
11		eqeq2 2633	. . . . 5 |- ( y = 4 -> ( ( 3 + 1 ) = y <-> ( 3 + 1 ) = 4 ) )
12	10, 11	3anbi23d 1402	. . . 4 |- ( y = 4 -> ( ( 3 e. CC /\ y e. CC /\ ( 3 + 1 ) = y ) <-> ( 3 e. CC /\ 4 e. CC /\ ( 3 + 1 ) = 4 ) ) )
13		eqid 2622	. . . 4 |- { <. x , y >. | ( x e. CC /\ y e. CC /\ ( x + 1 ) = y ) } = { <. x , y >. | ( x e. CC /\ y e. CC /\ ( x + 1 ) = y ) }
14	4, 5, 9, 12, 13	brab 4998	. . 3 |- ( 3 { <. x , y >. | ( x e. CC /\ y e. CC /\ ( x + 1 ) = y ) } 4 <-> ( 3 e. CC /\ 4 e. CC /\ ( 3 + 1 ) = 4 ) )
15	1, 2, 3, 14	mpbir3an 1244	. 2 |- 3 { <. x , y >. | ( x e. CC /\ y e. CC /\ ( x + 1 ) = y ) } 4
16		breq 4655	. 2 |- ( R = { <. x , y >. | ( x e. CC /\ y e. CC /\ ( x + 1 ) = y ) } -> ( 3 R 4 <-> 3 { <. x , y >. | ( x e. CC /\ y e. CC /\ ( x + 1 ) = y ) } 4 ) )
17	15, 16	mpbiri 248	1 |- ( R = { <. x , y >. | ( x e. CC /\ y e. CC /\ ( x + 1 ) = y ) } -> 3 R 4 )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   {copab 4712  (class class class)co 6650   CCcc 9934   1c1 9937   + caddc 9939   3c3 11071   4c4 11072

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  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-i2m1 10004  ax-1ne0 10005  ax-rrecex 10008  ax-cnre 10009

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-uni 4437  df-br 4654  df-opab 4713  df-iota 5851  df-fv 5896  df-ov 6653  df-2 11079  df-3 11080  df-4 11081

This theorem is referenced by: (None)