Theorem inl11 6950

Description: Left injection is one-to-one. (Contributed by Jim Kingdon, 12-Jul-2023.)

Assertion

Ref	Expression
inl11	|- ( ( A e. V /\ B e. W ) -> ( (inl ` A ) = (inl ` B ) <-> A = B ) )

Proof of Theorem inl11

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-inl 6932	. . . 4 |- inl = ( x e. _V |-> <. (/) , x >. )
2		opeq2 3706	. . . 4 |- ( x = A -> <. (/) , x >. = <. (/) , A >. )
3		elex 2697	. . . . 5 |- ( A e. V -> A e. _V )
4	3	adantr 274	. . . 4 |- ( ( A e. V /\ B e. W ) -> A e. _V )
5		0ex 4055	. . . . 5 |- (/) e. _V
6		simpl 108	. . . . 5 |- ( ( A e. V /\ B e. W ) -> A e. V )
7		opexg 4150	. . . . 5 |- ( ( (/) e. _V /\ A e. V ) -> <. (/) , A >. e. _V )
8	5, 6, 7	sylancr 410	. . . 4 |- ( ( A e. V /\ B e. W ) -> <. (/) , A >. e. _V )
9	1, 2, 4, 8	fvmptd3 5514	. . 3 |- ( ( A e. V /\ B e. W ) -> (inl ` A ) = <. (/) , A >. )
10		opeq2 3706	. . . 4 |- ( x = B -> <. (/) , x >. = <. (/) , B >. )
11		elex 2697	. . . . 5 |- ( B e. W -> B e. _V )
12	11	adantl 275	. . . 4 |- ( ( A e. V /\ B e. W ) -> B e. _V )
13	5	a1i 9	. . . . 5 |- ( ( A e. V /\ B e. W ) -> (/) e. _V )
14		opexg 4150	. . . . 5 |- ( ( (/) e. _V /\ B e. W ) -> <. (/) , B >. e. _V )
15	13, 14	sylancom 416	. . . 4 |- ( ( A e. V /\ B e. W ) -> <. (/) , B >. e. _V )
16	1, 10, 12, 15	fvmptd3 5514	. . 3 |- ( ( A e. V /\ B e. W ) -> (inl ` B ) = <. (/) , B >. )
17	9, 16	eqeq12d 2154	. 2 |- ( ( A e. V /\ B e. W ) -> ( (inl ` A ) = (inl ` B ) <-> <. (/) , A >. = <. (/) , B >. ) )
18		opthg 4160	. . . . 5 |- ( ( (/) e. _V /\ A e. V ) -> ( <. (/) , A >. = <. (/) , B >. <-> ( (/) = (/) /\ A = B ) ) )
19	5, 18	mpan 420	. . . 4 |- ( A e. V -> ( <. (/) , A >. = <. (/) , B >. <-> ( (/) = (/) /\ A = B ) ) )
20		eqid 2139	. . . . 5 |- (/) = (/)
21	20	biantrur 301	. . . 4 |- ( A = B <-> ( (/) = (/) /\ A = B ) )
22	19, 21	syl6bbr 197	. . 3 |- ( A e. V -> ( <. (/) , A >. = <. (/) , B >. <-> A = B ) )
23	22	adantr 274	. 2 |- ( ( A e. V /\ B e. W ) -> ( <. (/) , A >. = <. (/) , B >. <-> A = B ) )
24	17, 23	bitrd 187	1 |- ( ( A e. V /\ B e. W ) -> ( (inl ` A ) = (inl ` B ) <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   _Vcvv 2686   (/)c0 3363   <.cop 3530   ` cfv 5123  inlcinl 6930

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-in1 603  ax-in2 604  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-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131

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-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  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-iota 5088  df-fun 5125  df-fv 5131  df-inl 6932

This theorem is referenced by:  omp1eomlem  6979  difinfsnlem  6984