Theorem f1opr 33519

Description: Condition for an operation to be one-to-one. (Contributed by Jeff Madsen, 17-Jun-2010.)

Assertion

Ref	Expression
f1opr	|- ( F : ( A X. B ) -1-1-> C <-> ( F : ( A X. B ) --> C /\ A. r e. A A. s e. B A. t e. A A. u e. B ( ( r F s ) = ( t F u ) -> ( r = t /\ s = u ) ) ) )

Distinct variable groups:   A, r, s, t, u   B, r, s, t, u   F, r, s, t, u

Allowed substitution hints:   C( u, t, s, r)

Proof of Theorem f1opr

Dummy variables v w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dff13 6512	. 2 |- ( F : ( A X. B ) -1-1-> C <-> ( F : ( A X. B ) --> C /\ A. v e. ( A X. B ) A. w e. ( A X. B ) ( ( F ` v ) = ( F ` w ) -> v = w ) ) )
2		fveq2 6191	. . . . . . . . 9 |- ( v = <. r , s >. -> ( F ` v ) = ( F ` <. r , s >. ) )
3		df-ov 6653	. . . . . . . . 9 |- ( r F s ) = ( F ` <. r , s >. )
4	2, 3	syl6eqr 2674	. . . . . . . 8 |- ( v = <. r , s >. -> ( F ` v ) = ( r F s ) )
5	4	eqeq1d 2624	. . . . . . 7 |- ( v = <. r , s >. -> ( ( F ` v ) = ( F ` w ) <-> ( r F s ) = ( F ` w ) ) )
6		eqeq1 2626	. . . . . . 7 |- ( v = <. r , s >. -> ( v = w <-> <. r , s >. = w ) )
7	5, 6	imbi12d 334	. . . . . 6 |- ( v = <. r , s >. -> ( ( ( F ` v ) = ( F ` w ) -> v = w ) <-> ( ( r F s ) = ( F ` w ) -> <. r , s >. = w ) ) )
8	7	ralbidv 2986	. . . . 5 |- ( v = <. r , s >. -> ( A. w e. ( A X. B ) ( ( F ` v ) = ( F ` w ) -> v = w ) <-> A. w e. ( A X. B ) ( ( r F s ) = ( F ` w ) -> <. r , s >. = w ) ) )
9	8	ralxp 5263	. . . 4 |- ( A. v e. ( A X. B ) A. w e. ( A X. B ) ( ( F ` v ) = ( F ` w ) -> v = w ) <-> A. r e. A A. s e. B A. w e. ( A X. B ) ( ( r F s ) = ( F ` w ) -> <. r , s >. = w ) )
10		fveq2 6191	. . . . . . . . 9 |- ( w = <. t , u >. -> ( F ` w ) = ( F ` <. t , u >. ) )
11		df-ov 6653	. . . . . . . . 9 |- ( t F u ) = ( F ` <. t , u >. )
12	10, 11	syl6eqr 2674	. . . . . . . 8 |- ( w = <. t , u >. -> ( F ` w ) = ( t F u ) )
13	12	eqeq2d 2632	. . . . . . 7 |- ( w = <. t , u >. -> ( ( r F s ) = ( F ` w ) <-> ( r F s ) = ( t F u ) ) )
14		eqeq2 2633	. . . . . . . 8 |- ( w = <. t , u >. -> ( <. r , s >. = w <-> <. r , s >. = <. t , u >. ) )
15		vex 3203	. . . . . . . . 9 |- r e. _V
16		vex 3203	. . . . . . . . 9 |- s e. _V
17	15, 16	opth 4945	. . . . . . . 8 |- ( <. r , s >. = <. t , u >. <-> ( r = t /\ s = u ) )
18	14, 17	syl6bb 276	. . . . . . 7 |- ( w = <. t , u >. -> ( <. r , s >. = w <-> ( r = t /\ s = u ) ) )
19	13, 18	imbi12d 334	. . . . . 6 |- ( w = <. t , u >. -> ( ( ( r F s ) = ( F ` w ) -> <. r , s >. = w ) <-> ( ( r F s ) = ( t F u ) -> ( r = t /\ s = u ) ) ) )
20	19	ralxp 5263	. . . . 5 |- ( A. w e. ( A X. B ) ( ( r F s ) = ( F ` w ) -> <. r , s >. = w ) <-> A. t e. A A. u e. B ( ( r F s ) = ( t F u ) -> ( r = t /\ s = u ) ) )
21	20	2ralbii 2981	. . . 4 |- ( A. r e. A A. s e. B A. w e. ( A X. B ) ( ( r F s ) = ( F ` w ) -> <. r , s >. = w ) <-> A. r e. A A. s e. B A. t e. A A. u e. B ( ( r F s ) = ( t F u ) -> ( r = t /\ s = u ) ) )
22	9, 21	bitri 264	. . 3 |- ( A. v e. ( A X. B ) A. w e. ( A X. B ) ( ( F ` v ) = ( F ` w ) -> v = w ) <-> A. r e. A A. s e. B A. t e. A A. u e. B ( ( r F s ) = ( t F u ) -> ( r = t /\ s = u ) ) )
23	22	anbi2i 730	. 2 |- ( ( F : ( A X. B ) --> C /\ A. v e. ( A X. B ) A. w e. ( A X. B ) ( ( F ` v ) = ( F ` w ) -> v = w ) ) <-> ( F : ( A X. B ) --> C /\ A. r e. A A. s e. B A. t e. A A. u e. B ( ( r F s ) = ( t F u ) -> ( r = t /\ s = u ) ) ) )
24	1, 23	bitri 264	1 |- ( F : ( A X. B ) -1-1-> C <-> ( F : ( A X. B ) --> C /\ A. r e. A A. s e. B A. t e. A A. u e. B ( ( r F s ) = ( t F u ) -> ( r = t /\ s = u ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   A.wral 2912   <.cop 4183   X. cxp 5112   -->wf 5884   -1-1->wf1 5885   ` cfv 5888  (class class class)co 6650

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  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-iun 4522  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fv 5896  df-ov 6653

This theorem is referenced by: (None)