Theorem opiota 7229

Description: The property of a uniquely specified ordered pair. The proof uses properties of the iota description binder. (Contributed by Mario Carneiro, 21-May-2015.)

Hypotheses

Ref	Expression
opiota.1	|- I = ( iota z E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) )
opiota.2	|- X = ( 1st ` I )
opiota.3	|- Y = ( 2nd ` I )
opiota.4	|- ( x = C -> ( ph <-> ps ) )
opiota.5	|- ( y = D -> ( ps <-> ch ) )

Assertion

Ref	Expression
opiota	|- ( E! z E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) -> ( ( C e. A /\ D e. B /\ ch ) <-> ( C = X /\ D = Y ) ) )

Distinct variable groups:   x, y, z, A   x, B, y, z   x, C, y, z   ch, y   ph, z   x, D, y, z   ps, x

Allowed substitution hints:   ph( x, y)   ps( y, z)   ch( x, z)   I( x, y, z)   X( x, y, z)   Y( x, y, z)

Proof of Theorem opiota

Step	Hyp	Ref	Expression
1		opiota.4	. . . . . . 7 |- ( x = C -> ( ph <-> ps ) )
2		opiota.5	. . . . . . 7 |- ( y = D -> ( ps <-> ch ) )
3	1, 2	ceqsrex2v 3338	. . . . . 6 |- ( ( C e. A /\ D e. B ) -> ( E. x e. A E. y e. B ( ( x = C /\ y = D ) /\ ph ) <-> ch ) )
4	3	bicomd 213	. . . . 5 |- ( ( C e. A /\ D e. B ) -> ( ch <-> E. x e. A E. y e. B ( ( x = C /\ y = D ) /\ ph ) ) )
5		opex 4932	. . . . . . . 8 |- <. C , D >. e. _V
6	5	a1i 11	. . . . . . 7 |- ( E! z E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) -> <. C , D >. e. _V )
7		id 22	. . . . . . 7 |- ( E! z E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) -> E! z E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) )
8		eqeq1 2626	. . . . . . . . . . 11 |- ( z = <. C , D >. -> ( z = <. x , y >. <-> <. C , D >. = <. x , y >. ) )
9		eqcom 2629	. . . . . . . . . . . 12 |- ( <. C , D >. = <. x , y >. <-> <. x , y >. = <. C , D >. )
10		vex 3203	. . . . . . . . . . . . 13 |- x e. _V
11		vex 3203	. . . . . . . . . . . . 13 |- y e. _V
12	10, 11	opth 4945	. . . . . . . . . . . 12 |- ( <. x , y >. = <. C , D >. <-> ( x = C /\ y = D ) )
13	9, 12	bitri 264	. . . . . . . . . . 11 |- ( <. C , D >. = <. x , y >. <-> ( x = C /\ y = D ) )
14	8, 13	syl6bb 276	. . . . . . . . . 10 |- ( z = <. C , D >. -> ( z = <. x , y >. <-> ( x = C /\ y = D ) ) )
15	14	anbi1d 741	. . . . . . . . 9 |- ( z = <. C , D >. -> ( ( z = <. x , y >. /\ ph ) <-> ( ( x = C /\ y = D ) /\ ph ) ) )
16	15	2rexbidv 3057	. . . . . . . 8 |- ( z = <. C , D >. -> ( E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) <-> E. x e. A E. y e. B ( ( x = C /\ y = D ) /\ ph ) ) )
17	16	adantl 482	. . . . . . 7 |- ( ( E! z E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) /\ z = <. C , D >. ) -> ( E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) <-> E. x e. A E. y e. B ( ( x = C /\ y = D ) /\ ph ) ) )
18		nfeu1 2480	. . . . . . 7 |- F/ z E! z E. x e. A E. y e. B ( z = <. x , y >. /\ ph )
19		nfvd 1844	. . . . . . 7 |- ( E! z E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) -> F/ z E. x e. A E. y e. B ( ( x = C /\ y = D ) /\ ph ) )
20		nfcvd 2765	. . . . . . 7 |- ( E! z E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) -> F/_ z <. C , D >. )
21	6, 7, 17, 18, 19, 20	iota2df 5875	. . . . . 6 |- ( E! z E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) -> ( E. x e. A E. y e. B ( ( x = C /\ y = D ) /\ ph ) <-> ( iota z E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) ) = <. C , D >. ) )
22		eqcom 2629	. . . . . . 7 |- ( <. C , D >. = I <-> I = <. C , D >. )
23		opiota.1	. . . . . . . 8 |- I = ( iota z E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) )
24	23	eqeq1i 2627	. . . . . . 7 |- ( I = <. C , D >. <-> ( iota z E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) ) = <. C , D >. )
25	22, 24	bitri 264	. . . . . 6 |- ( <. C , D >. = I <-> ( iota z E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) ) = <. C , D >. )
26	21, 25	syl6bbr 278	. . . . 5 |- ( E! z E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) -> ( E. x e. A E. y e. B ( ( x = C /\ y = D ) /\ ph ) <-> <. C , D >. = I ) )
27	4, 26	sylan9bbr 737	. . . 4 |- ( ( E! z E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) /\ ( C e. A /\ D e. B ) ) -> ( ch <-> <. C , D >. = I ) )
28	27	pm5.32da 673	. . 3 |- ( E! z E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) -> ( ( ( C e. A /\ D e. B ) /\ ch ) <-> ( ( C e. A /\ D e. B ) /\ <. C , D >. = I ) ) )
29		opelxpi 5148	. . . . . . . . . 10 |- ( ( x e. A /\ y e. B ) -> <. x , y >. e. ( A X. B ) )
30		simpl 473	. . . . . . . . . . 11 |- ( ( z = <. x , y >. /\ ph ) -> z = <. x , y >. )
31	30	eleq1d 2686	. . . . . . . . . 10 |- ( ( z = <. x , y >. /\ ph ) -> ( z e. ( A X. B ) <-> <. x , y >. e. ( A X. B ) ) )
32	29, 31	syl5ibrcom 237	. . . . . . . . 9 |- ( ( x e. A /\ y e. B ) -> ( ( z = <. x , y >. /\ ph ) -> z e. ( A X. B ) ) )
33	32	rexlimivv 3036	. . . . . . . 8 |- ( E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) -> z e. ( A X. B ) )
34	33	abssi 3677	. . . . . . 7 |- { z | E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) } C_ ( A X. B )
35		iotacl 5874	. . . . . . 7 |- ( E! z E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) -> ( iota z E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) ) e. { z | E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) } )
36	34, 35	sseldi 3601	. . . . . 6 |- ( E! z E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) -> ( iota z E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) ) e. ( A X. B ) )
37	23, 36	syl5eqel 2705	. . . . 5 |- ( E! z E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) -> I e. ( A X. B ) )
38		opelxp 5146	. . . . . 6 |- ( <. C , D >. e. ( A X. B ) <-> ( C e. A /\ D e. B ) )
39		eleq1 2689	. . . . . 6 |- ( <. C , D >. = I -> ( <. C , D >. e. ( A X. B ) <-> I e. ( A X. B ) ) )
40	38, 39	syl5bbr 274	. . . . 5 |- ( <. C , D >. = I -> ( ( C e. A /\ D e. B ) <-> I e. ( A X. B ) ) )
41	37, 40	syl5ibrcom 237	. . . 4 |- ( E! z E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) -> ( <. C , D >. = I -> ( C e. A /\ D e. B ) ) )
42	41	pm4.71rd 667	. . 3 |- ( E! z E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) -> ( <. C , D >. = I <-> ( ( C e. A /\ D e. B ) /\ <. C , D >. = I ) ) )
43		1st2nd2 7205	. . . . 5 |- ( I e. ( A X. B ) -> I = <. ( 1st ` I ) , ( 2nd ` I ) >. )
44	37, 43	syl 17	. . . 4 |- ( E! z E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) -> I = <. ( 1st ` I ) , ( 2nd ` I ) >. )
45	44	eqeq2d 2632	. . 3 |- ( E! z E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) -> ( <. C , D >. = I <-> <. C , D >. = <. ( 1st ` I ) , ( 2nd ` I ) >. ) )
46	28, 42, 45	3bitr2d 296	. 2 |- ( E! z E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) -> ( ( ( C e. A /\ D e. B ) /\ ch ) <-> <. C , D >. = <. ( 1st ` I ) , ( 2nd ` I ) >. ) )
47		df-3an 1039	. 2 |- ( ( C e. A /\ D e. B /\ ch ) <-> ( ( C e. A /\ D e. B ) /\ ch ) )
48		opiota.2	. . . . 5 |- X = ( 1st ` I )
49	48	eqeq2i 2634	. . . 4 |- ( C = X <-> C = ( 1st ` I ) )
50		opiota.3	. . . . 5 |- Y = ( 2nd ` I )
51	50	eqeq2i 2634	. . . 4 |- ( D = Y <-> D = ( 2nd ` I ) )
52	49, 51	anbi12i 733	. . 3 |- ( ( C = X /\ D = Y ) <-> ( C = ( 1st ` I ) /\ D = ( 2nd ` I ) ) )
53		fvex 6201	. . . 4 |- ( 1st ` I ) e. _V
54		fvex 6201	. . . 4 |- ( 2nd ` I ) e. _V
55	53, 54	opth2 4949	. . 3 |- ( <. C , D >. = <. ( 1st ` I ) , ( 2nd ` I ) >. <-> ( C = ( 1st ` I ) /\ D = ( 2nd ` I ) ) )
56	52, 55	bitr4i 267	. 2 |- ( ( C = X /\ D = Y ) <-> <. C , D >. = <. ( 1st ` I ) , ( 2nd ` I ) >. )
57	46, 47, 56	3bitr4g 303	1 |- ( E! z E. x e. A E. y e. B ( z = <. x , y >. /\ ph ) -> ( ( C e. A /\ D e. B /\ ch ) <-> ( C = X /\ D = Y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E!weu 2470   {cab 2608   E.wrex 2913   _Vcvv 3200   <.cop 4183   X. cxp 5112   iotacio 5849   ` cfv 5888   1stc1st 7166   2ndc2nd 7167

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-pow 4843  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-sbc 3436  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-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by:  oeeui  7682