Theorem inl11 7030

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 7012	. . . 4 |- inl = ( x e. _V |-> <. (/) , x >. )
2		opeq2 3759	. . . 4 |- ( x = A -> <. (/) , x >. = <. (/) , A >. )
3		elex 2737	. . . . 5 |- ( A e. V -> A e. _V )
4	3	adantr 274	. . . 4 |- ( ( A e. V /\ B e. W ) -> A e. _V )
5		0ex 4109	. . . . 5 |- (/) e. _V
6		simpl 108	. . . . 5 |- ( ( A e. V /\ B e. W ) -> A e. V )
7		opexg 4206	. . . . 5 |- ( ( (/) e. _V /\ A e. V ) -> <. (/) , A >. e. _V )
8	5, 6, 7	sylancr 411	. . . 4 |- ( ( A e. V /\ B e. W ) -> <. (/) , A >. e. _V )
9	1, 2, 4, 8	fvmptd3 5579	. . 3 |- ( ( A e. V /\ B e. W ) -> (inl ` A ) = <. (/) , A >. )
10		opeq2 3759	. . . 4 |- ( x = B -> <. (/) , x >. = <. (/) , B >. )
11		elex 2737	. . . . 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 4206	. . . . 5 |- ( ( (/) e. _V /\ B e. W ) -> <. (/) , B >. e. _V )
15	13, 14	sylancom 417	. . . 4 |- ( ( A e. V /\ B e. W ) -> <. (/) , B >. e. _V )
16	1, 10, 12, 15	fvmptd3 5579	. . 3 |- ( ( A e. V /\ B e. W ) -> (inl ` B ) = <. (/) , B >. )
17	9, 16	eqeq12d 2180	. 2 |- ( ( A e. V /\ B e. W ) -> ( (inl ` A ) = (inl ` B ) <-> <. (/) , A >. = <. (/) , B >. ) )
18		opthg 4216	. . . . 5 |- ( ( (/) e. _V /\ A e. V ) -> ( <. (/) , A >. = <. (/) , B >. <-> ( (/) = (/) /\ A = B ) ) )
19	5, 18	mpan 421	. . . 4 |- ( A e. V -> ( <. (/) , A >. = <. (/) , B >. <-> ( (/) = (/) /\ A = B ) ) )
20		eqid 2165	. . . . 5 |- (/) = (/)
21	20	biantrur 301	. . . 4 |- ( A = B <-> ( (/) = (/) /\ A = B ) )
22	19, 21	bitr4di 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 1343   e. wcel 2136   _Vcvv 2726   (/)c0 3409   <.cop 3579   ` cfv 5188  inlcinl 7010

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-inl 7012

This theorem is referenced by:  omp1eomlem  7059  difinfsnlem  7064