Theorem upxp 12441

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 5645	. . . 4 |- ( A e. D -> ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) e. _V )
2		eueq 2855	. . . 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 1002	. 2 |- ( ( A e. D /\ F : A --> B /\ G : A --> C ) -> E! h h = ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) )
5		ffn 5272	. . . . . . . 8 |- ( h : A --> ( B X. C ) -> h Fn A )
6	5	3ad2ant1 1002	. . . . . . 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 5553	. . . . . . . . . . . . 13 |- ( ( F : A --> B /\ x e. A ) -> ( F ` x ) e. B )
9		ffvelrn 5553	. . . . . . . . . . . . 13 |- ( ( G : A --> C /\ x e. A ) -> ( G ` x ) e. C )
10		opelxpi 4571	. . . . . . . . . . . . 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 582	. . . . . . . . . . 11 |- ( ( ( F : A --> B /\ G : A --> C ) /\ x e. A ) -> <. ( F ` x ) , ( G ` x ) >. e. ( B X. C ) )
13	12	ralrimiva 2505	. . . . . . . . . 10 |- ( ( F : A --> B /\ G : A --> C ) -> A. x e. A <. ( F ` x ) , ( G ` x ) >. e. ( B X. C ) )
14	13	3adant1 999	. . . . . . . . 9 |- ( ( A e. D /\ F : A --> B /\ G : A --> C ) -> A. x e. A <. ( F ` x ) , ( G ` x ) >. e. ( B X. C ) )
15		eqid 2139	. . . . . . . . . 10 |- ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) = ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. )
16	15	fmpt 5570	. . . . . . . . 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 5273	. . . . . . 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 4647	. . . . . . . . . . 11 |- ( B X. C ) C_ ( _V X. _V )
21		ffvelrn 5553	. . . . . . . . . . 11 |- ( ( h : A --> ( B X. C ) /\ z e. A ) -> ( h ` z ) e. ( B X. C ) )
22	20, 21	sseldi 3095	. . . . . . . . . 10 |- ( ( h : A --> ( B X. C ) /\ z e. A ) -> ( h ` z ) e. ( _V X. _V ) )
23	22	3ad2antl1 1143	. . . . . . . . 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 467	. . . . . . . 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 5420	. . . . . . . . . . . 12 |- ( F = ( P o. h ) -> ( F ` z ) = ( ( P o. h ) ` z ) )
26		upxp.1	. . . . . . . . . . . . . 14 |- P = ( 1st |` ( B X. C ) )
27	26	coeq1i 4698	. . . . . . . . . . . . 13 |- ( P o. h ) = ( ( 1st |` ( B X. C ) ) o. h )
28	27	fveq1i 5422	. . . . . . . . . . . 12 |- ( ( P o. h ) ` z ) = ( ( ( 1st |` ( B X. C ) ) o. h ) ` z )
29	25, 28	syl6eq 2188	. . . . . . . . . . 11 |- ( F = ( P o. h ) -> ( F ` z ) = ( ( ( 1st |` ( B X. C ) ) o. h ) ` z ) )
30	29	3ad2ant2 1003	. . . . . . . . . 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 480	. . . . . . . . 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 987	. . . . . . . . . 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 5492	. . . . . . . . . 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 1143	. . . . . . . . . . 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 467	. . . . . . . . . 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 5446	. . . . . . . . 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 2177	. . . . . . . 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 5420	. . . . . . . . . . . 12 |- ( G = ( Q o. h ) -> ( G ` z ) = ( ( Q o. h ) ` z ) )
40		upxp.2	. . . . . . . . . . . . . 14 |- Q = ( 2nd |` ( B X. C ) )
41	40	coeq1i 4698	. . . . . . . . . . . . 13 |- ( Q o. h ) = ( ( 2nd |` ( B X. C ) ) o. h )
42	41	fveq1i 5422	. . . . . . . . . . . 12 |- ( ( Q o. h ) ` z ) = ( ( ( 2nd |` ( B X. C ) ) o. h ) ` z )
43	39, 42	syl6eq 2188	. . . . . . . . . . 11 |- ( G = ( Q o. h ) -> ( G ` z ) = ( ( ( 2nd |` ( B X. C ) ) o. h ) ` z ) )
44	43	3ad2ant3 1004	. . . . . . . . . 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 480	. . . . . . . . 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 5492	. . . . . . . . . 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 5446	. . . . . . . . 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 2177	. . . . . . . 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 6070	. . . . . . . 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 1214	. . . . . . 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 5421	. . . . . . . . 9 |- ( x = z -> ( F ` x ) = ( F ` z ) )
53		fveq2 5421	. . . . . . . . 9 |- ( x = z -> ( G ` x ) = ( G ` z ) )
54	52, 53	opeq12d 3713	. . . . . . . 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 2217	. . . . . . . 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 5514	. . . . . . 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 2175	. . . . . 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 5521	. . . . 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 5272	. . . . . . . . 9 |- ( F : A --> B -> F Fn A )
62	61	3ad2ant2 1003	. . . . . . . 8 |- ( ( A e. D /\ F : A --> B /\ G : A --> C ) -> F Fn A )
63		fo1st 6055	. . . . . . . . . . 11 |- 1st : _V -onto-> _V
64		fofn 5347	. . . . . . . . . . 11 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
65	63, 64	ax-mp 5	. . . . . . . . . 10 |- 1st Fn _V
66		ssv 3119	. . . . . . . . . 10 |- ( B X. C ) C_ _V
67		fnssres 5236	. . . . . . . . . 10 |- ( ( 1st Fn _V /\ ( B X. C ) C_ _V ) -> ( 1st |` ( B X. C ) ) Fn ( B X. C ) )
68	65, 66, 67	mp2an 422	. . . . . . . . 9 |- ( 1st |` ( B X. C ) ) Fn ( B X. C )
69	17	frnd 5282	. . . . . . . . 9 |- ( ( A e. D /\ F : A --> B /\ G : A --> C ) -> ran ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) C_ ( B X. C ) )
70		fnco 5231	. . . . . . . . 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 1320	. . . . . . . 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 5492	. . . . . . . . . 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 985	. . . . . . . . . . . . 13 |- ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ z e. A ) -> F : A --> B )
76	75, 74	ffvelrnd 5556	. . . . . . . . . . . 12 |- ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ z e. A ) -> ( F ` z ) e. B )
77		simpl3 986	. . . . . . . . . . . . 13 |- ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ z e. A ) -> G : A --> C )
78	77, 74	ffvelrnd 5556	. . . . . . . . . . . 12 |- ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ z e. A ) -> ( G ` z ) e. C )
79	76, 78	opelxpd 4572	. . . . . . . . . . 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 5514	. . . . . . . . . 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 5425	. . . . . . . . 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 5553	. . . . . . . . . . . . . 14 |- ( ( F : A --> B /\ z e. A ) -> ( F ` z ) e. B )
83		ffvelrn 5553	. . . . . . . . . . . . . 14 |- ( ( G : A --> C /\ z e. A ) -> ( G ` z ) e. C )
84		opelxpi 4571	. . . . . . . . . . . . . 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 582	. . . . . . . . . . . 12 |- ( ( ( F : A --> B /\ G : A --> C ) /\ z e. A ) -> <. ( F ` z ) , ( G ` z ) >. e. ( B X. C ) )
87	86	3adantl1 1137	. . . . . . . . . . 11 |- ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ z e. A ) -> <. ( F ` z ) , ( G ` z ) >. e. ( B X. C ) )
88	87	fvresd 5446	. . . . . . . . . 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 6048	. . . . . . . . . . 11 |- ( ( ( F ` z ) e. B /\ ( G ` z ) e. C ) -> ( 1st ` <. ( F ` z ) , ( G ` z ) >. ) = ( F ` z ) )
90	76, 78, 89	syl2anc 408	. . . . . . . . . 10 |- ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ z e. A ) -> ( 1st ` <. ( F ` z ) , ( G ` z ) >. ) = ( F ` z ) )
91	88, 90	eqtrd 2172	. . . . . . . . 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 2177	. . . . . . . 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 5521	. . . . . . 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 4698	. . . . . . 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	syl6eqr 2190	. . . . . 6 |- ( ( A e. D /\ F : A --> B /\ G : A --> C ) -> F = ( P o. ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ) )
96		ffn 5272	. . . . . . . . 9 |- ( G : A --> C -> G Fn A )
97	96	3ad2ant3 1004	. . . . . . . 8 |- ( ( A e. D /\ F : A --> B /\ G : A --> C ) -> G Fn A )
98		fo2nd 6056	. . . . . . . . . . 11 |- 2nd : _V -onto-> _V
99		fofn 5347	. . . . . . . . . . 11 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
100	98, 99	ax-mp 5	. . . . . . . . . 10 |- 2nd Fn _V
101		fnssres 5236	. . . . . . . . . 10 |- ( ( 2nd Fn _V /\ ( B X. C ) C_ _V ) -> ( 2nd |` ( B X. C ) ) Fn ( B X. C ) )
102	100, 66, 101	mp2an 422	. . . . . . . . 9 |- ( 2nd |` ( B X. C ) ) Fn ( B X. C )
103		fnco 5231	. . . . . . . . 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 1320	. . . . . . . 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 5492	. . . . . . . . . 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 5425	. . . . . . . . 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 5446	. . . . . . . . . 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 6049	. . . . . . . . . . 11 |- ( ( ( F ` z ) e. B /\ ( G ` z ) e. C ) -> ( 2nd ` <. ( F ` z ) , ( G ` z ) >. ) = ( G ` z ) )
110	76, 78, 109	syl2anc 408	. . . . . . . . . 10 |- ( ( ( A e. D /\ F : A --> B /\ G : A --> C ) /\ z e. A ) -> ( 2nd ` <. ( F ` z ) , ( G ` z ) >. ) = ( G ` z ) )
111	108, 110	eqtrd 2172	. . . . . . . . 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 2177	. . . . . . . 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 5521	. . . . . . 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 4698	. . . . . . 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	syl6eqr 2190	. . . . . 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 1161	. . . . 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 5255	. . . . . 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 4697	. . . . . . 7 |- ( h = ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) -> ( P o. h ) = ( P o. ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ) )
119	118	eqeq2d 2151	. . . . . 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 4697	. . . . . . 7 |- ( h = ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) -> ( Q o. h ) = ( Q o. ( x e. A |-> <. ( F ` x ) , ( G ` x ) >. ) ) )
121	120	eqeq2d 2151	. . . . . 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 1290	. . . . 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 2007	. 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 962   = wceq 1331   e. wcel 1480   E!weu 1999   A.wral 2416   _Vcvv 2686   C_ wss 3071   <.cop 3530   |-> cmpt 3989   X. cxp 4537   ran crn 4540   |` cres 4541   o. ccom 4543   Fn wfn 5118   -->wf 5119   -onto->wfo 5121   ` cfv 5123   1stc1st 6036   2ndc2nd 6037

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1st 6038  df-2nd 6039

This theorem is referenced by:  uptx  12443  txcn  12444