Theorem foeqcnvco 6555

Description: Condition for function equality in terms of vanishing of the composition with the converse. EDITORIAL: Is there a relation-algebraic proof of this? (Contributed by Stefan O'Rear, 12-Feb-2015.)

Assertion

Ref	Expression
foeqcnvco	|- ( ( F : A -onto-> B /\ G : A -onto-> B ) -> ( F = G <-> ( F o. `' G ) = ( _I |` B ) ) )

Proof of Theorem foeqcnvco

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

Step	Hyp	Ref	Expression
1		fococnv2 6162	. . . 4 |- ( F : A -onto-> B -> ( F o. `' F ) = ( _I |` B ) )
2		cnveq 5296	. . . . . 6 |- ( F = G -> `' F = `' G )
3	2	coeq2d 5284	. . . . 5 |- ( F = G -> ( F o. `' F ) = ( F o. `' G ) )
4	3	eqeq1d 2624	. . . 4 |- ( F = G -> ( ( F o. `' F ) = ( _I |` B ) <-> ( F o. `' G ) = ( _I |` B ) ) )
5	1, 4	syl5ibcom 235	. . 3 |- ( F : A -onto-> B -> ( F = G -> ( F o. `' G ) = ( _I |` B ) ) )
6	5	adantr 481	. 2 |- ( ( F : A -onto-> B /\ G : A -onto-> B ) -> ( F = G -> ( F o. `' G ) = ( _I |` B ) ) )
7		fofn 6117	. . . . 5 |- ( F : A -onto-> B -> F Fn A )
8	7	ad2antrr 762	. . . 4 |- ( ( ( F : A -onto-> B /\ G : A -onto-> B ) /\ ( F o. `' G ) = ( _I |` B ) ) -> F Fn A )
9		fofn 6117	. . . . 5 |- ( G : A -onto-> B -> G Fn A )
10	9	ad2antlr 763	. . . 4 |- ( ( ( F : A -onto-> B /\ G : A -onto-> B ) /\ ( F o. `' G ) = ( _I |` B ) ) -> G Fn A )
11	9	adantl 482	. . . . . . . . . . . 12 |- ( ( F : A -onto-> B /\ G : A -onto-> B ) -> G Fn A )
12		fnopfv 6351	. . . . . . . . . . . 12 |- ( ( G Fn A /\ x e. A ) -> <. x , ( G ` x ) >. e. G )
13	11, 12	sylan 488	. . . . . . . . . . 11 |- ( ( ( F : A -onto-> B /\ G : A -onto-> B ) /\ x e. A ) -> <. x , ( G ` x ) >. e. G )
14		fvex 6201	. . . . . . . . . . . . 13 |- ( G ` x ) e. _V
15		vex 3203	. . . . . . . . . . . . 13 |- x e. _V
16	14, 15	brcnv 5305	. . . . . . . . . . . 12 |- ( ( G ` x ) `' G x <-> x G ( G ` x ) )
17		df-br 4654	. . . . . . . . . . . 12 |- ( x G ( G ` x ) <-> <. x , ( G ` x ) >. e. G )
18	16, 17	bitri 264	. . . . . . . . . . 11 |- ( ( G ` x ) `' G x <-> <. x , ( G ` x ) >. e. G )
19	13, 18	sylibr 224	. . . . . . . . . 10 |- ( ( ( F : A -onto-> B /\ G : A -onto-> B ) /\ x e. A ) -> ( G ` x ) `' G x )
20	7	adantr 481	. . . . . . . . . . . 12 |- ( ( F : A -onto-> B /\ G : A -onto-> B ) -> F Fn A )
21		fnopfv 6351	. . . . . . . . . . . 12 |- ( ( F Fn A /\ x e. A ) -> <. x , ( F ` x ) >. e. F )
22	20, 21	sylan 488	. . . . . . . . . . 11 |- ( ( ( F : A -onto-> B /\ G : A -onto-> B ) /\ x e. A ) -> <. x , ( F ` x ) >. e. F )
23		df-br 4654	. . . . . . . . . . 11 |- ( x F ( F ` x ) <-> <. x , ( F ` x ) >. e. F )
24	22, 23	sylibr 224	. . . . . . . . . 10 |- ( ( ( F : A -onto-> B /\ G : A -onto-> B ) /\ x e. A ) -> x F ( F ` x ) )
25		breq2 4657	. . . . . . . . . . . 12 |- ( y = x -> ( ( G ` x ) `' G y <-> ( G ` x ) `' G x ) )
26		breq1 4656	. . . . . . . . . . . 12 |- ( y = x -> ( y F ( F ` x ) <-> x F ( F ` x ) ) )
27	25, 26	anbi12d 747	. . . . . . . . . . 11 |- ( y = x -> ( ( ( G ` x ) `' G y /\ y F ( F ` x ) ) <-> ( ( G ` x ) `' G x /\ x F ( F ` x ) ) ) )
28	15, 27	spcev 3300	. . . . . . . . . 10 |- ( ( ( G ` x ) `' G x /\ x F ( F ` x ) ) -> E. y ( ( G ` x ) `' G y /\ y F ( F ` x ) ) )
29	19, 24, 28	syl2anc 693	. . . . . . . . 9 |- ( ( ( F : A -onto-> B /\ G : A -onto-> B ) /\ x e. A ) -> E. y ( ( G ` x ) `' G y /\ y F ( F ` x ) ) )
30		fvex 6201	. . . . . . . . . 10 |- ( F ` x ) e. _V
31	14, 30	brco 5292	. . . . . . . . 9 |- ( ( G ` x ) ( F o. `' G ) ( F ` x ) <-> E. y ( ( G ` x ) `' G y /\ y F ( F ` x ) ) )
32	29, 31	sylibr 224	. . . . . . . 8 |- ( ( ( F : A -onto-> B /\ G : A -onto-> B ) /\ x e. A ) -> ( G ` x ) ( F o. `' G ) ( F ` x ) )
33	32	adantlr 751	. . . . . . 7 |- ( ( ( ( F : A -onto-> B /\ G : A -onto-> B ) /\ ( F o. `' G ) = ( _I |` B ) ) /\ x e. A ) -> ( G ` x ) ( F o. `' G ) ( F ` x ) )
34		breq 4655	. . . . . . . 8 |- ( ( F o. `' G ) = ( _I |` B ) -> ( ( G ` x ) ( F o. `' G ) ( F ` x ) <-> ( G ` x ) ( _I |` B ) ( F ` x ) ) )
35	34	ad2antlr 763	. . . . . . 7 |- ( ( ( ( F : A -onto-> B /\ G : A -onto-> B ) /\ ( F o. `' G ) = ( _I |` B ) ) /\ x e. A ) -> ( ( G ` x ) ( F o. `' G ) ( F ` x ) <-> ( G ` x ) ( _I |` B ) ( F ` x ) ) )
36	33, 35	mpbid 222	. . . . . 6 |- ( ( ( ( F : A -onto-> B /\ G : A -onto-> B ) /\ ( F o. `' G ) = ( _I |` B ) ) /\ x e. A ) -> ( G ` x ) ( _I |` B ) ( F ` x ) )
37		fof 6115	. . . . . . . . . 10 |- ( G : A -onto-> B -> G : A --> B )
38	37	adantl 482	. . . . . . . . 9 |- ( ( F : A -onto-> B /\ G : A -onto-> B ) -> G : A --> B )
39	38	ffvelrnda 6359	. . . . . . . 8 |- ( ( ( F : A -onto-> B /\ G : A -onto-> B ) /\ x e. A ) -> ( G ` x ) e. B )
40		fof 6115	. . . . . . . . . 10 |- ( F : A -onto-> B -> F : A --> B )
41	40	adantr 481	. . . . . . . . 9 |- ( ( F : A -onto-> B /\ G : A -onto-> B ) -> F : A --> B )
42	41	ffvelrnda 6359	. . . . . . . 8 |- ( ( ( F : A -onto-> B /\ G : A -onto-> B ) /\ x e. A ) -> ( F ` x ) e. B )
43		resieq 5407	. . . . . . . 8 |- ( ( ( G ` x ) e. B /\ ( F ` x ) e. B ) -> ( ( G ` x ) ( _I |` B ) ( F ` x ) <-> ( G ` x ) = ( F ` x ) ) )
44	39, 42, 43	syl2anc 693	. . . . . . 7 |- ( ( ( F : A -onto-> B /\ G : A -onto-> B ) /\ x e. A ) -> ( ( G ` x ) ( _I |` B ) ( F ` x ) <-> ( G ` x ) = ( F ` x ) ) )
45	44	adantlr 751	. . . . . 6 |- ( ( ( ( F : A -onto-> B /\ G : A -onto-> B ) /\ ( F o. `' G ) = ( _I |` B ) ) /\ x e. A ) -> ( ( G ` x ) ( _I |` B ) ( F ` x ) <-> ( G ` x ) = ( F ` x ) ) )
46	36, 45	mpbid 222	. . . . 5 |- ( ( ( ( F : A -onto-> B /\ G : A -onto-> B ) /\ ( F o. `' G ) = ( _I |` B ) ) /\ x e. A ) -> ( G ` x ) = ( F ` x ) )
47	46	eqcomd 2628	. . . 4 |- ( ( ( ( F : A -onto-> B /\ G : A -onto-> B ) /\ ( F o. `' G ) = ( _I |` B ) ) /\ x e. A ) -> ( F ` x ) = ( G ` x ) )
48	8, 10, 47	eqfnfvd 6314	. . 3 |- ( ( ( F : A -onto-> B /\ G : A -onto-> B ) /\ ( F o. `' G ) = ( _I |` B ) ) -> F = G )
49	48	ex 450	. 2 |- ( ( F : A -onto-> B /\ G : A -onto-> B ) -> ( ( F o. `' G ) = ( _I |` B ) -> F = G ) )
50	6, 49	impbid 202	1 |- ( ( F : A -onto-> B /\ G : A -onto-> B ) -> ( F = G <-> ( F o. `' G ) = ( _I |` B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   <.cop 4183   class class class wbr 4653   _I cid 5023   `'ccnv 5113   |` cres 5116   o. ccom 5118   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   ` 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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896

This theorem is referenced by: (None)