Theorem bj-inftyexpiinj 33096

Description: Injectivity of the parameterization inftyexpi. Remark: a more conceptual proof would use bj-inftyexpiinv 33095 and the fact that a function with a retraction is injective. (Contributed by BJ, 22-Jun-2019.)

Assertion

Ref	Expression
bj-inftyexpiinj	|- ( ( A e. ( -u pi (,] pi ) /\ B e. ( -u pi (,] pi ) ) -> ( A = B <-> (inftyexpi ` A ) = (inftyexpi ` B ) ) )

Proof of Theorem bj-inftyexpiinj

Step	Hyp	Ref	Expression
1		fveq2 6191	. 2 |- ( A = B -> (inftyexpi ` A ) = (inftyexpi ` B ) )
2		fveq2 6191	. . 3 |- ( (inftyexpi ` A ) = (inftyexpi ` B ) -> ( 1st ` (inftyexpi ` A ) ) = ( 1st ` (inftyexpi ` B ) ) )
3		bj-inftyexpiinv 33095	. . . . . . 7 |- ( A e. ( -u pi (,] pi ) -> ( 1st ` (inftyexpi ` A ) ) = A )
4	3	adantr 481	. . . . . 6 |- ( ( A e. ( -u pi (,] pi ) /\ B e. ( -u pi (,] pi ) ) -> ( 1st ` (inftyexpi ` A ) ) = A )
5	4	eqeq1d 2624	. . . . 5 |- ( ( A e. ( -u pi (,] pi ) /\ B e. ( -u pi (,] pi ) ) -> ( ( 1st ` (inftyexpi ` A ) ) = ( 1st ` (inftyexpi ` B ) ) <-> A = ( 1st ` (inftyexpi ` B ) ) ) )
6	5	biimpd 219	. . . 4 |- ( ( A e. ( -u pi (,] pi ) /\ B e. ( -u pi (,] pi ) ) -> ( ( 1st ` (inftyexpi ` A ) ) = ( 1st ` (inftyexpi ` B ) ) -> A = ( 1st ` (inftyexpi ` B ) ) ) )
7		bj-inftyexpiinv 33095	. . . . . 6 |- ( B e. ( -u pi (,] pi ) -> ( 1st ` (inftyexpi ` B ) ) = B )
8	7	adantl 482	. . . . 5 |- ( ( A e. ( -u pi (,] pi ) /\ B e. ( -u pi (,] pi ) ) -> ( 1st ` (inftyexpi ` B ) ) = B )
9	8	eqeq2d 2632	. . . 4 |- ( ( A e. ( -u pi (,] pi ) /\ B e. ( -u pi (,] pi ) ) -> ( A = ( 1st ` (inftyexpi ` B ) ) <-> A = B ) )
10	6, 9	sylibd 229	. . 3 |- ( ( A e. ( -u pi (,] pi ) /\ B e. ( -u pi (,] pi ) ) -> ( ( 1st ` (inftyexpi ` A ) ) = ( 1st ` (inftyexpi ` B ) ) -> A = B ) )
11	2, 10	syl5 34	. 2 |- ( ( A e. ( -u pi (,] pi ) /\ B e. ( -u pi (,] pi ) ) -> ( (inftyexpi ` A ) = (inftyexpi ` B ) -> A = B ) )
12	1, 11	impbid2 216	1 |- ( ( A e. ( -u pi (,] pi ) /\ B e. ( -u pi (,] pi ) ) -> ( A = B <-> (inftyexpi ` A ) = (inftyexpi ` B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   1stc1st 7166   -ucneg 10267   (,]cioc 12176   picpi 14797  inftyexpi cinftyexpi 33093

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  ax-un 6949  ax-cnex 9992

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-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-iota 5851  df-fun 5890  df-fv 5896  df-1st 7168  df-bj-inftyexpi 33094

This theorem is referenced by:  bj-pinftynminfty  33114