Theorem fvcofneq 6367

Description: The values of two function compositions are equal if the values of the composed functions are pairwise equal. (Contributed by AV, 26-Jan-2019.)

Assertion

Ref	Expression
fvcofneq	|- ( ( G Fn A /\ K Fn B ) -> ( ( X e. ( A i^i B ) /\ ( G ` X ) = ( K ` X ) /\ A. x e. ( ran G i^i ran K ) ( F ` x ) = ( H ` x ) ) -> ( ( F o. G ) ` X ) = ( ( H o. K ) ` X ) ) )

Distinct variable groups:   x, F   x, G   x, H   x, K   x, X

Allowed substitution hints:   A( x)   B( x)

Proof of Theorem fvcofneq

Step	Hyp	Ref	Expression
1		simpl 473	. . . 4 |- ( ( G Fn A /\ K Fn B ) -> G Fn A )
2		elin 3796	. . . . . 6 |- ( X e. ( A i^i B ) <-> ( X e. A /\ X e. B ) )
3		simpl 473	. . . . . 6 |- ( ( X e. A /\ X e. B ) -> X e. A )
4	2, 3	sylbi 207	. . . . 5 |- ( X e. ( A i^i B ) -> X e. A )
5	4	3ad2ant1 1082	. . . 4 |- ( ( X e. ( A i^i B ) /\ ( G ` X ) = ( K ` X ) /\ A. x e. ( ran G i^i ran K ) ( F ` x ) = ( H ` x ) ) -> X e. A )
6		fvco2 6273	. . . 4 |- ( ( G Fn A /\ X e. A ) -> ( ( F o. G ) ` X ) = ( F ` ( G ` X ) ) )
7	1, 5, 6	syl2an 494	. . 3 |- ( ( ( G Fn A /\ K Fn B ) /\ ( X e. ( A i^i B ) /\ ( G ` X ) = ( K ` X ) /\ A. x e. ( ran G i^i ran K ) ( F ` x ) = ( H ` x ) ) ) -> ( ( F o. G ) ` X ) = ( F ` ( G ` X ) ) )
8		simpr 477	. . . . 5 |- ( ( G Fn A /\ K Fn B ) -> K Fn B )
9		simpr 477	. . . . . . 7 |- ( ( X e. A /\ X e. B ) -> X e. B )
10	2, 9	sylbi 207	. . . . . 6 |- ( X e. ( A i^i B ) -> X e. B )
11	10	3ad2ant1 1082	. . . . 5 |- ( ( X e. ( A i^i B ) /\ ( G ` X ) = ( K ` X ) /\ A. x e. ( ran G i^i ran K ) ( F ` x ) = ( H ` x ) ) -> X e. B )
12		fvco2 6273	. . . . 5 |- ( ( K Fn B /\ X e. B ) -> ( ( H o. K ) ` X ) = ( H ` ( K ` X ) ) )
13	8, 11, 12	syl2an 494	. . . 4 |- ( ( ( G Fn A /\ K Fn B ) /\ ( X e. ( A i^i B ) /\ ( G ` X ) = ( K ` X ) /\ A. x e. ( ran G i^i ran K ) ( F ` x ) = ( H ` x ) ) ) -> ( ( H o. K ) ` X ) = ( H ` ( K ` X ) ) )
14		fveq2 6191	. . . . . . 7 |- ( ( K ` X ) = ( G ` X ) -> ( H ` ( K ` X ) ) = ( H ` ( G ` X ) ) )
15	14	eqcoms 2630	. . . . . 6 |- ( ( G ` X ) = ( K ` X ) -> ( H ` ( K ` X ) ) = ( H ` ( G ` X ) ) )
16	15	3ad2ant2 1083	. . . . 5 |- ( ( X e. ( A i^i B ) /\ ( G ` X ) = ( K ` X ) /\ A. x e. ( ran G i^i ran K ) ( F ` x ) = ( H ` x ) ) -> ( H ` ( K ` X ) ) = ( H ` ( G ` X ) ) )
17	16	adantl 482	. . . 4 |- ( ( ( G Fn A /\ K Fn B ) /\ ( X e. ( A i^i B ) /\ ( G ` X ) = ( K ` X ) /\ A. x e. ( ran G i^i ran K ) ( F ` x ) = ( H ` x ) ) ) -> ( H ` ( K ` X ) ) = ( H ` ( G ` X ) ) )
18		id 22	. . . . . . . . . . . 12 |- ( G Fn A -> G Fn A )
19		fnfvelrn 6356	. . . . . . . . . . . 12 |- ( ( G Fn A /\ X e. A ) -> ( G ` X ) e. ran G )
20	18, 4, 19	syl2anr 495	. . . . . . . . . . 11 |- ( ( X e. ( A i^i B ) /\ G Fn A ) -> ( G ` X ) e. ran G )
21	20	ex 450	. . . . . . . . . 10 |- ( X e. ( A i^i B ) -> ( G Fn A -> ( G ` X ) e. ran G ) )
22		id 22	. . . . . . . . . . . 12 |- ( K Fn B -> K Fn B )
23		fnfvelrn 6356	. . . . . . . . . . . 12 |- ( ( K Fn B /\ X e. B ) -> ( K ` X ) e. ran K )
24	22, 10, 23	syl2anr 495	. . . . . . . . . . 11 |- ( ( X e. ( A i^i B ) /\ K Fn B ) -> ( K ` X ) e. ran K )
25	24	ex 450	. . . . . . . . . 10 |- ( X e. ( A i^i B ) -> ( K Fn B -> ( K ` X ) e. ran K ) )
26	21, 25	anim12d 586	. . . . . . . . 9 |- ( X e. ( A i^i B ) -> ( ( G Fn A /\ K Fn B ) -> ( ( G ` X ) e. ran G /\ ( K ` X ) e. ran K ) ) )
27		eleq1 2689	. . . . . . . . . . . 12 |- ( ( K ` X ) = ( G ` X ) -> ( ( K ` X ) e. ran K <-> ( G ` X ) e. ran K ) )
28	27	eqcoms 2630	. . . . . . . . . . 11 |- ( ( G ` X ) = ( K ` X ) -> ( ( K ` X ) e. ran K <-> ( G ` X ) e. ran K ) )
29	28	anbi2d 740	. . . . . . . . . 10 |- ( ( G ` X ) = ( K ` X ) -> ( ( ( G ` X ) e. ran G /\ ( K ` X ) e. ran K ) <-> ( ( G ` X ) e. ran G /\ ( G ` X ) e. ran K ) ) )
30		elin 3796	. . . . . . . . . . 11 |- ( ( G ` X ) e. ( ran G i^i ran K ) <-> ( ( G ` X ) e. ran G /\ ( G ` X ) e. ran K ) )
31	30	biimpri 218	. . . . . . . . . 10 |- ( ( ( G ` X ) e. ran G /\ ( G ` X ) e. ran K ) -> ( G ` X ) e. ( ran G i^i ran K ) )
32	29, 31	syl6bi 243	. . . . . . . . 9 |- ( ( G ` X ) = ( K ` X ) -> ( ( ( G ` X ) e. ran G /\ ( K ` X ) e. ran K ) -> ( G ` X ) e. ( ran G i^i ran K ) ) )
33	26, 32	sylan9 689	. . . . . . . 8 |- ( ( X e. ( A i^i B ) /\ ( G ` X ) = ( K ` X ) ) -> ( ( G Fn A /\ K Fn B ) -> ( G ` X ) e. ( ran G i^i ran K ) ) )
34		fveq2 6191	. . . . . . . . . . . 12 |- ( x = ( G ` X ) -> ( F ` x ) = ( F ` ( G ` X ) ) )
35		fveq2 6191	. . . . . . . . . . . 12 |- ( x = ( G ` X ) -> ( H ` x ) = ( H ` ( G ` X ) ) )
36	34, 35	eqeq12d 2637	. . . . . . . . . . 11 |- ( x = ( G ` X ) -> ( ( F ` x ) = ( H ` x ) <-> ( F ` ( G ` X ) ) = ( H ` ( G ` X ) ) ) )
37	36	rspcva 3307	. . . . . . . . . 10 |- ( ( ( G ` X ) e. ( ran G i^i ran K ) /\ A. x e. ( ran G i^i ran K ) ( F ` x ) = ( H ` x ) ) -> ( F ` ( G ` X ) ) = ( H ` ( G ` X ) ) )
38	37	eqcomd 2628	. . . . . . . . 9 |- ( ( ( G ` X ) e. ( ran G i^i ran K ) /\ A. x e. ( ran G i^i ran K ) ( F ` x ) = ( H ` x ) ) -> ( H ` ( G ` X ) ) = ( F ` ( G ` X ) ) )
39	38	ex 450	. . . . . . . 8 |- ( ( G ` X ) e. ( ran G i^i ran K ) -> ( A. x e. ( ran G i^i ran K ) ( F ` x ) = ( H ` x ) -> ( H ` ( G ` X ) ) = ( F ` ( G ` X ) ) ) )
40	33, 39	syl6 35	. . . . . . 7 |- ( ( X e. ( A i^i B ) /\ ( G ` X ) = ( K ` X ) ) -> ( ( G Fn A /\ K Fn B ) -> ( A. x e. ( ran G i^i ran K ) ( F ` x ) = ( H ` x ) -> ( H ` ( G ` X ) ) = ( F ` ( G ` X ) ) ) ) )
41	40	com23 86	. . . . . 6 |- ( ( X e. ( A i^i B ) /\ ( G ` X ) = ( K ` X ) ) -> ( A. x e. ( ran G i^i ran K ) ( F ` x ) = ( H ` x ) -> ( ( G Fn A /\ K Fn B ) -> ( H ` ( G ` X ) ) = ( F ` ( G ` X ) ) ) ) )
42	41	3impia 1261	. . . . 5 |- ( ( X e. ( A i^i B ) /\ ( G ` X ) = ( K ` X ) /\ A. x e. ( ran G i^i ran K ) ( F ` x ) = ( H ` x ) ) -> ( ( G Fn A /\ K Fn B ) -> ( H ` ( G ` X ) ) = ( F ` ( G ` X ) ) ) )
43	42	impcom 446	. . . 4 |- ( ( ( G Fn A /\ K Fn B ) /\ ( X e. ( A i^i B ) /\ ( G ` X ) = ( K ` X ) /\ A. x e. ( ran G i^i ran K ) ( F ` x ) = ( H ` x ) ) ) -> ( H ` ( G ` X ) ) = ( F ` ( G ` X ) ) )
44	13, 17, 43	3eqtrrd 2661	. . 3 |- ( ( ( G Fn A /\ K Fn B ) /\ ( X e. ( A i^i B ) /\ ( G ` X ) = ( K ` X ) /\ A. x e. ( ran G i^i ran K ) ( F ` x ) = ( H ` x ) ) ) -> ( F ` ( G ` X ) ) = ( ( H o. K ) ` X ) )
45	7, 44	eqtrd 2656	. 2 |- ( ( ( G Fn A /\ K Fn B ) /\ ( X e. ( A i^i B ) /\ ( G ` X ) = ( K ` X ) /\ A. x e. ( ran G i^i ran K ) ( F ` x ) = ( H ` x ) ) ) -> ( ( F o. G ) ` X ) = ( ( H o. K ) ` X ) )
46	45	ex 450	1 |- ( ( G Fn A /\ K Fn B ) -> ( ( X e. ( A i^i B ) /\ ( G ` X ) = ( K ` X ) /\ A. x e. ( ran G i^i ran K ) ( F ` x ) = ( H ` x ) ) -> ( ( F o. G ) ` X ) = ( ( H o. K ) ` X ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   i^i cin 3573   ran crn 5115   o. ccom 5118   Fn wfn 5883   ` cfv 5888

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

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-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-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  fvcosymgeq  17849