Theorem upxp 12912

Description: Universal property of the Cartesian product considered as a categorical product in the category of sets. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)

Hypotheses

Ref	Expression
upxp.1	|- P = ( 1st |` ( B X. C ) )
upxp.2	|- Q = ( 2nd |` ( B X. C ) )

Assertion

Ref	Expression
upxp	|- ( ( A e. D /\ F : A --> B /\ G : A --> C ) -> E! h ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) )

Distinct variable groups:   A, h   B, h   C, h   h, F   h, G   D, h

Allowed substitution hints:   P( h)   Q( h)

Proof of Theorem upxp

Dummy variables x z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mptexg 5710	. . . 4 |- ( A e. D -> ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) e. _V )
2		eueq 2897	. . . 4 |- ( ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) e. _V <-> E! h h = ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) )
3	1, 2	sylib 121	. . 3 |- ( A e. D -> E! h h = ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) )
4	3	3ad2ant1 1008	. 2 |- ( ( A e. D /\ F : A --> B /\ G : A --> C ) -> E! h h = ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) )
5		ffn 5337	. . . . . . . 8 |- ( h : A --> ( B X. C ) -> h Fn A )
6	5	3ad2ant1 1008	. . . . . . 7 |- ( ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) -> h Fn A )
7	6	adantl 275	. . . . . 6 |- ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) ) -> h Fn A )
8		ffvelrn 5618	. . . . . . . . . . . . 13 |- ( ( F : A --> B /\ x e. A ) -> ( F ` x ) e. B )
9		ffvelrn 5618	. . . . . . . . . . . . 13 |- ( ( G : A --> C /\ x e. A ) -> ( G ` x ) e. C )
10		opelxpi 4636	. . . . . . . . . . . . 13 |- ( ( ( F ` x ) e. B /\ ( G ` x ) e. C ) -> <. ( F ` x ) , ( G ` x ) >. e. ( B X. C ) )
11	8, 9, 10	syl2an 287	. . . . . . . . . . . 12 |- ( ( ( F : A --> B /\ x e. A ) /\ ( G : A --> C /\ x e. A ) ) -> <. ( F ` x ) , ( G ` x ) >. e. ( B X. C ) )
12	11	anandirs 583	. . . . . . . . . . 11 |- ( ( ( F : A --> B /\ G : A --> C ) /\ x e. A ) -> <. ( F ` x ) , ( G ` x ) >. e. ( B X. C ) )
13	12	ralrimiva 2539	. . . . . . . . . 10 |- ( ( F : A --> B /\ G : A --> C ) -> A. x e. A <. ( F ` x ) , ( G ` x ) >. e. ( B X. C ) )
14	13	3adant1 1005	. . . . . . . . 9 |- ( ( A e. D /\ F : A --> B /\ G : A --> C ) -> A. x e. A <. ( F ` x ) , ( G ` x ) >. e. ( B X. C ) )
15		eqid 2165	. . . . . . . . . 10 |- ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) = ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. )
16	15	fmpt 5635	. . . . . . . . 9 |- ( A. x e. A <. ( F ` x ) , ( G ` x ) >. e. ( B X. C ) <-> ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) : A --> ( B X. C ) )
17	14, 16	sylib 121	. . . . . . . 8 |- ( ( A e. D /\ F : A --> B /\ G : A --> C ) -> ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) : A --> ( B X. C ) )
18	17	ffnd 5338	. . . . . . 7 |- ( ( A e. D /\ F : A --> B /\ G : A --> C ) -> ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) Fn A )
19	18	adantr 274	. . . . . 6 |- ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) ) -> ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) Fn A )
20		xpss 4712	. . . . . . . . . . 11 |- ( B X. C ) C_ ( _V X. _V )
21		ffvelrn 5618	. . . . . . . . . . 11 |- ( ( h : A --> ( B X. C ) /\ z e. A ) -> ( h ` z ) e. ( B X. C ) )
22	20, 21	sselid 3140	. . . . . . . . . 10 |- ( ( h : A --> ( B X. C ) /\ z e. A ) -> ( h ` z ) e. ( _V X. _V ) )
23	22	3ad2antl1 1149	. . . . . . . . 9 |- ( ( ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) /\ z e. A ) -> ( h ` z ) e. ( _V X. _V ) )
24	23	adantll 468	. . . . . . . 8 |- ( ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) ) /\ z e. A ) -> ( h ` z ) e. ( _V X. _V ) )
25		fveq1 5485	. . . . . . . . . . . 12 |- ( F = ( P o. h ) -> ( F ` z ) = ( ( P o. h ) ` z ) )
26		upxp.1	. . . . . . . . . . . . . 14 |- P = ( 1st |` ( B X. C ) )
27	26	coeq1i 4763	. . . . . . . . . . . . 13 |- ( P o. h ) = ( ( 1st |` ( B X. C ) ) o. h )
28	27	fveq1i 5487	. . . . . . . . . . . 12 |- ( ( P o. h ) ` z ) = ( ( ( 1st |` ( B X. C ) ) o. h ) ` z )
29	25, 28	eqtrdi 2215	. . . . . . . . . . 11 |- ( F = ( P o. h ) -> ( F ` z ) = ( ( ( 1st |` ( B X. C ) ) o. h ) ` z ) )
30	29	3ad2ant2 1009	. . . . . . . . . 10 |- ( ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) -> ( F ` z ) = ( ( ( 1st |` ( B X. C ) ) o. h ) ` z ) )
31	30	ad2antlr 481	. . . . . . . . 9 |- ( ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) ) /\ z e. A ) -> ( F ` z ) = ( ( ( 1st |` ( B X. C ) ) o. h ) ` z ) )
32		simpr1 993	. . . . . . . . . 10 |- ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) ) -> h : A --> ( B X. C ) )
33		fvco3 5557	. . . . . . . . . 10 |- ( ( h : A --> ( B X. C ) /\ z e. A ) -> ( ( ( 1st |` ( B X. C ) ) o. h ) ` z ) = ( ( 1st |` ( B X. C ) ) ` ( h ` z ) ) )
34	32, 33	sylan 281	. . . . . . . . 9 |- ( ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) ) /\ z e. A ) -> ( ( ( 1st |` ( B X. C ) ) o. h ) ` z ) = ( ( 1st |` ( B X. C ) ) ` ( h ` z ) ) )
35	21	3ad2antl1 1149	. . . . . . . . . . 11 |- ( ( ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) /\ z e. A ) -> ( h ` z ) e. ( B X. C ) )
36	35	adantll 468	. . . . . . . . . 10 |- ( ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) ) /\ z e. A ) -> ( h ` z ) e. ( B X. C ) )
37	36	fvresd 5511	. . . . . . . . 9 |- ( ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) ) /\ z e. A ) -> ( ( 1st |` ( B X. C ) ) ` ( h ` z ) ) = ( 1st ` ( h ` z ) ) )
38	31, 34, 37	3eqtrrd 2203	. . . . . . . 8 |- ( ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) ) /\ z e. A ) -> ( 1st ` ( h ` z ) ) = ( F ` z ) )
39		fveq1 5485	. . . . . . . . . . . 12 |- ( G = ( Q o. h ) -> ( G ` z ) = ( ( Q o. h ) ` z ) )
40		upxp.2	. . . . . . . . . . . . . 14 |- Q = ( 2nd |` ( B X. C ) )
41	40	coeq1i 4763	. . . . . . . . . . . . 13 |- ( Q o. h ) = ( ( 2nd |` ( B X. C ) ) o. h )
42	41	fveq1i 5487	. . . . . . . . . . . 12 |- ( ( Q o. h ) ` z ) = ( ( ( 2nd |` ( B X. C ) ) o. h ) ` z )
43	39, 42	eqtrdi 2215	. . . . . . . . . . 11 |- ( G = ( Q o. h ) -> ( G ` z ) = ( ( ( 2nd |` ( B X. C ) ) o. h ) ` z ) )
44	43	3ad2ant3 1010	. . . . . . . . . 10 |- ( ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) -> ( G ` z ) = ( ( ( 2nd |` ( B X. C ) ) o. h ) ` z ) )
45	44	ad2antlr 481	. . . . . . . . 9 |- ( ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) ) /\ z e. A ) -> ( G ` z ) = ( ( ( 2nd |` ( B X. C ) ) o. h ) ` z ) )
46		fvco3 5557	. . . . . . . . . 10 |- ( ( h : A --> ( B X. C ) /\ z e. A ) -> ( ( ( 2nd |` ( B X. C ) ) o. h ) ` z ) = ( ( 2nd |` ( B X. C ) ) ` ( h ` z ) ) )
47	32, 46	sylan 281	. . . . . . . . 9 |- ( ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) ) /\ z e. A ) -> ( ( ( 2nd |` ( B X. C ) ) o. h ) ` z ) = ( ( 2nd |` ( B X. C ) ) ` ( h ` z ) ) )
48	36	fvresd 5511	. . . . . . . . 9 |- ( ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) ) /\ z e. A ) -> ( ( 2nd |` ( B X. C ) ) ` ( h ` z ) ) = ( 2nd ` ( h ` z ) ) )
49	45, 47, 48	3eqtrrd 2203	. . . . . . . 8 |- ( ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) ) /\ z e. A ) -> ( 2nd ` ( h ` z ) ) = ( G ` z ) )
50		eqopi 6140	. . . . . . . 8 |- ( ( ( h ` z ) e. ( _V X. _V ) /\ ( ( 1st ` ( h ` z ) ) = ( F ` z ) /\ ( 2nd ` ( h ` z ) ) = ( G ` z ) ) ) -> ( h ` z ) = <. ( F ` z ) , ( G ` z ) >. )
51	24, 38, 49, 50	syl12anc 1226	. . . . . . 7 |- ( ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) ) /\ z e. A ) -> ( h ` z ) = <. ( F ` z ) , ( G ` z ) >. )
52		fveq2 5486	. . . . . . . . 9 |- ( x = z -> ( F ` x ) = ( F ` z ) )
53		fveq2 5486	. . . . . . . . 9 |- ( x = z -> ( G ` x ) = ( G ` z ) )
54	52, 53	opeq12d 3766	. . . . . . . 8 |- ( x = z -> <. ( F ` x ) , ( G ` x ) >. = <. ( F ` z ) , ( G ` z ) >. )
55		simpr 109	. . . . . . . 8 |- ( ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) ) /\ z e. A ) -> z e. A )
56	51, 36	eqeltrrd 2244	. . . . . . . 8 |- ( ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) ) /\ z e. A ) -> <. ( F ` z ) , ( G ` z ) >. e. ( B X. C ) )
57	15, 54, 55, 56	fvmptd3 5579	. . . . . . 7 |- ( ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) ) /\ z e. A ) -> ( ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ` z ) = <. ( F ` z ) , ( G ` z ) >. )
58	51, 57	eqtr4d 2201	. . . . . 6 |- ( ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) ) /\ z e. A ) -> ( h ` z ) = ( ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ` z ) )
59	7, 19, 58	eqfnfvd 5586	. . . . 5 |- ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) ) -> h = ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) )
60	59	ex 114	. . . 4 |- ( ( A e. D /\ F : A --> B /\ G : A --> C ) -> ( ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) -> h = ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ) )
61		ffn 5337	. . . . . . . . 9 |- ( F : A --> B -> F Fn A )
62	61	3ad2ant2 1009	. . . . . . . 8 |- ( ( A e. D /\ F : A --> B /\ G : A --> C ) -> F Fn A )
63		fo1st 6125	. . . . . . . . . . 11 |- 1st : _V -onto-> _V
64		fofn 5412	. . . . . . . . . . 11 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
65	63, 64	ax-mp 5	. . . . . . . . . 10 |- 1st Fn _V
66		ssv 3164	. . . . . . . . . 10 |- ( B X. C ) C_ _V
67		fnssres 5301	. . . . . . . . . 10 |- ( ( 1st Fn _V /\ ( B X. C ) C_ _V ) -> ( 1st |` ( B X. C ) ) Fn ( B X. C ) )
68	65, 66, 67	mp2an 423	. . . . . . . . 9 |- ( 1st |` ( B X. C ) ) Fn ( B X. C )
69	17	frnd 5347	. . . . . . . . 9 |- ( ( A e. D /\ F : A --> B /\ G : A --> C ) -> ran ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) C_ ( B X. C ) )
70		fnco 5296	. . . . . . . . 9 |- ( ( ( 1st |` ( B X. C ) ) Fn ( B X. C ) /\ ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) Fn A /\ ran ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) C_ ( B X. C ) ) -> ( ( 1st |` ( B X. C ) ) o. ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ) Fn A )
71	68, 18, 69, 70	mp3an2i 1332	. . . . . . . 8 |- ( ( A e. D /\ F : A --> B /\ G : A --> C ) -> ( ( 1st |` ( B X. C ) ) o. ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ) Fn A )
72		fvco3 5557	. . . . . . . . . 10 |- ( ( ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) : A --> ( B X. C ) /\ z e. A ) -> ( ( ( 1st |` ( B X. C ) ) o. ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ) ` z ) = ( ( 1st |` ( B X. C ) ) ` ( ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ` z ) ) )
73	17, 72	sylan 281	. . . . . . . . 9 |- ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ z e. A ) -> ( ( ( 1st |` ( B X. C ) ) o. ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ) ` z ) = ( ( 1st |` ( B X. C ) ) ` ( ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ` z ) ) )
74		simpr 109	. . . . . . . . . . 11 |- ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ z e. A ) -> z e. A )
75		simpl2 991	. . . . . . . . . . . . 13 |- ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ z e. A ) -> F : A --> B )
76	75, 74	ffvelrnd 5621	. . . . . . . . . . . 12 |- ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ z e. A ) -> ( F ` z ) e. B )
77		simpl3 992	. . . . . . . . . . . . 13 |- ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ z e. A ) -> G : A --> C )
78	77, 74	ffvelrnd 5621	. . . . . . . . . . . 12 |- ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ z e. A ) -> ( G ` z ) e. C )
79	76, 78	opelxpd 4637	. . . . . . . . . . 11 |- ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ z e. A ) -> <. ( F ` z ) , ( G ` z ) >. e. ( B X. C ) )
80	15, 54, 74, 79	fvmptd3 5579	. . . . . . . . . 10 |- ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ z e. A ) -> ( ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ` z ) = <. ( F ` z ) , ( G ` z ) >. )
81	80	fveq2d 5490	. . . . . . . . 9 |- ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ z e. A ) -> ( ( 1st |` ( B X. C ) ) ` ( ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ` z ) ) = ( ( 1st |` ( B X. C ) ) ` <. ( F ` z ) , ( G ` z ) >. ) )
82		ffvelrn 5618	. . . . . . . . . . . . . 14 |- ( ( F : A --> B /\ z e. A ) -> ( F ` z ) e. B )
83		ffvelrn 5618	. . . . . . . . . . . . . 14 |- ( ( G : A --> C /\ z e. A ) -> ( G ` z ) e. C )
84		opelxpi 4636	. . . . . . . . . . . . . 14 |- ( ( ( F ` z ) e. B /\ ( G ` z ) e. C ) -> <. ( F ` z ) , ( G ` z ) >. e. ( B X. C ) )
85	82, 83, 84	syl2an 287	. . . . . . . . . . . . 13 |- ( ( ( F : A --> B /\ z e. A ) /\ ( G : A --> C /\ z e. A ) ) -> <. ( F ` z ) , ( G ` z ) >. e. ( B X. C ) )
86	85	anandirs 583	. . . . . . . . . . . 12 |- ( ( ( F : A --> B /\ G : A --> C ) /\ z e. A ) -> <. ( F ` z ) , ( G ` z ) >. e. ( B X. C ) )
87	86	3adantl1 1143	. . . . . . . . . . 11 |- ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ z e. A ) -> <. ( F ` z ) , ( G ` z ) >. e. ( B X. C ) )
88	87	fvresd 5511	. . . . . . . . . 10 |- ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ z e. A ) -> ( ( 1st |` ( B X. C ) ) ` <. ( F ` z ) , ( G ` z ) >. ) = ( 1st ` <. ( F ` z ) , ( G ` z ) >. ) )
89		op1stg 6118	. . . . . . . . . . 11 |- ( ( ( F ` z ) e. B /\ ( G ` z ) e. C ) -> ( 1st ` <. ( F ` z ) , ( G ` z ) >. ) = ( F ` z ) )
90	76, 78, 89	syl2anc 409	. . . . . . . . . 10 |- ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ z e. A ) -> ( 1st ` <. ( F ` z ) , ( G ` z ) >. ) = ( F ` z ) )
91	88, 90	eqtrd 2198	. . . . . . . . 9 |- ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ z e. A ) -> ( ( 1st |` ( B X. C ) ) ` <. ( F ` z ) , ( G ` z ) >. ) = ( F ` z ) )
92	73, 81, 91	3eqtrrd 2203	. . . . . . . 8 |- ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ z e. A ) -> ( F ` z ) = ( ( ( 1st |` ( B X. C ) ) o. ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ) ` z ) )
93	62, 71, 92	eqfnfvd 5586	. . . . . . 7 |- ( ( A e. D /\ F : A --> B /\ G : A --> C ) -> F = ( ( 1st |` ( B X. C ) ) o. ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ) )
94	26	coeq1i 4763	. . . . . . 7 |- ( P o. ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ) = ( ( 1st |` ( B X. C ) ) o. ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) )
95	93, 94	eqtr4di 2217	. . . . . 6 |- ( ( A e. D /\ F : A --> B /\ G : A --> C ) -> F = ( P o. ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ) )
96		ffn 5337	. . . . . . . . 9 |- ( G : A --> C -> G Fn A )
97	96	3ad2ant3 1010	. . . . . . . 8 |- ( ( A e. D /\ F : A --> B /\ G : A --> C ) -> G Fn A )
98		fo2nd 6126	. . . . . . . . . . 11 |- 2nd : _V -onto-> _V
99		fofn 5412	. . . . . . . . . . 11 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
100	98, 99	ax-mp 5	. . . . . . . . . 10 |- 2nd Fn _V
101		fnssres 5301	. . . . . . . . . 10 |- ( ( 2nd Fn _V /\ ( B X. C ) C_ _V ) -> ( 2nd |` ( B X. C ) ) Fn ( B X. C ) )
102	100, 66, 101	mp2an 423	. . . . . . . . 9 |- ( 2nd |` ( B X. C ) ) Fn ( B X. C )
103		fnco 5296	. . . . . . . . 9 |- ( ( ( 2nd |` ( B X. C ) ) Fn ( B X. C ) /\ ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) Fn A /\ ran ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) C_ ( B X. C ) ) -> ( ( 2nd |` ( B X. C ) ) o. ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ) Fn A )
104	102, 18, 69, 103	mp3an2i 1332	. . . . . . . 8 |- ( ( A e. D /\ F : A --> B /\ G : A --> C ) -> ( ( 2nd |` ( B X. C ) ) o. ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ) Fn A )
105		fvco3 5557	. . . . . . . . . 10 |- ( ( ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) : A --> ( B X. C ) /\ z e. A ) -> ( ( ( 2nd |` ( B X. C ) ) o. ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ) ` z ) = ( ( 2nd |` ( B X. C ) ) ` ( ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ` z ) ) )
106	17, 105	sylan 281	. . . . . . . . 9 |- ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ z e. A ) -> ( ( ( 2nd |` ( B X. C ) ) o. ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ) ` z ) = ( ( 2nd |` ( B X. C ) ) ` ( ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ` z ) ) )
107	80	fveq2d 5490	. . . . . . . . 9 |- ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ z e. A ) -> ( ( 2nd |` ( B X. C ) ) ` ( ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ` z ) ) = ( ( 2nd |` ( B X. C ) ) ` <. ( F ` z ) , ( G ` z ) >. ) )
108	87	fvresd 5511	. . . . . . . . . 10 |- ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ z e. A ) -> ( ( 2nd |` ( B X. C ) ) ` <. ( F ` z ) , ( G ` z ) >. ) = ( 2nd ` <. ( F ` z ) , ( G ` z ) >. ) )
109		op2ndg 6119	. . . . . . . . . . 11 |- ( ( ( F ` z ) e. B /\ ( G ` z ) e. C ) -> ( 2nd ` <. ( F ` z ) , ( G ` z ) >. ) = ( G ` z ) )
110	76, 78, 109	syl2anc 409	. . . . . . . . . 10 |- ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ z e. A ) -> ( 2nd ` <. ( F ` z ) , ( G ` z ) >. ) = ( G ` z ) )
111	108, 110	eqtrd 2198	. . . . . . . . 9 |- ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ z e. A ) -> ( ( 2nd |` ( B X. C ) ) ` <. ( F ` z ) , ( G ` z ) >. ) = ( G ` z ) )
112	106, 107, 111	3eqtrrd 2203	. . . . . . . 8 |- ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ z e. A ) -> ( G ` z ) = ( ( ( 2nd |` ( B X. C ) ) o. ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ) ` z ) )
113	97, 104, 112	eqfnfvd 5586	. . . . . . 7 |- ( ( A e. D /\ F : A --> B /\ G : A --> C ) -> G = ( ( 2nd |` ( B X. C ) ) o. ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ) )
114	40	coeq1i 4763	. . . . . . 7 |- ( Q o. ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ) = ( ( 2nd |` ( B X. C ) ) o. ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) )
115	113, 114	eqtr4di 2217	. . . . . 6 |- ( ( A e. D /\ F : A --> B /\ G : A --> C ) -> G = ( Q o. ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ) )
116	17, 95, 115	3jca 1167	. . . . 5 |- ( ( A e. D /\ F : A --> B /\ G : A --> C ) -> ( ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) : A --> ( B X. C ) /\ F = ( P o. ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ) /\ G = ( Q o. ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ) ) )
117		feq1 5320	. . . . . 6 |- ( h = ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) -> ( h : A --> ( B X. C ) <-> ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) : A --> ( B X. C ) ) )
118		coeq2 4762	. . . . . . 7 |- ( h = ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) -> ( P o. h ) = ( P o. ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ) )
119	118	eqeq2d 2177	. . . . . 6 |- ( h = ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) -> ( F = ( P o. h ) <-> F = ( P o. ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ) ) )
120		coeq2 4762	. . . . . . 7 |- ( h = ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) -> ( Q o. h ) = ( Q o. ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ) )
121	120	eqeq2d 2177	. . . . . 6 |- ( h = ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) -> ( G = ( Q o. h ) <-> G = ( Q o. ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ) ) )
122	117, 119, 121	3anbi123d 1302	. . . . 5 |- ( h = ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) -> ( ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) <-> ( ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) : A --> ( B X. C ) /\ F = ( P o. ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ) /\ G = ( Q o. ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ) ) ) )
123	116, 122	syl5ibrcom 156	. . . 4 |- ( ( A e. D /\ F : A --> B /\ G : A --> C ) -> ( h = ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) -> ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) ) )
124	60, 123	impbid 128	. . 3 |- ( ( A e. D /\ F : A --> B /\ G : A --> C ) -> ( ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) <-> h = ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ) )
125	124	eubidv 2022	. 2 |- ( ( A e. D /\ F : A --> B /\ G : A --> C ) -> ( E! h ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) <-> E! h h = ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ) )
126	4, 125	mpbird 166	1 |- ( ( A e. D /\ F : A --> B /\ G : A --> C ) -> E! h ( h : A --> ( B X. C ) /\ F = ( P o. h ) /\ G = ( Q o. h ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968   = wceq 1343   E!weu 2014   e. wcel 2136   A.wral 2444   _Vcvv 2726   C_ wss 3116   <.cop 3579   |-> cmpt 4043   X. cxp 4602   ran crn 4605   |` cres 4606   o. ccom 4608   Fn wfn 5183   -->wf 5184   -onto->wfo 5186   ` cfv 5188   1stc1st 6106   2ndc2nd 6107

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-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-1st 6108  df-2nd 6109

This theorem is referenced by:  uptx  12914  txcn  12915