Theorem inl11 7124

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 7106	. . . 4 |- inl = ( x e. _V |-> <. (/) , x >. )
2		opeq2 3805	. . . 4 |- ( x = A -> <. (/) , x >. = <. (/) , A >. )
3		elex 2771	. . . . 5 |- ( A e. V -> A e. _V )
4	3	adantr 276	. . . 4 |- ( ( A e. V /\ B e. W ) -> A e. _V )
5		0ex 4156	. . . . 5 |- (/) e. _V
6		simpl 109	. . . . 5 |- ( ( A e. V /\ B e. W ) -> A e. V )
7		opexg 4257	. . . . 5 |- ( ( (/) e. _V /\ A e. V ) -> <. (/) , A >. e. _V )
8	5, 6, 7	sylancr 414	. . . 4 |- ( ( A e. V /\ B e. W ) -> <. (/) , A >. e. _V )
9	1, 2, 4, 8	fvmptd3 5651	. . 3 |- ( ( A e. V /\ B e. W ) -> (inl ` A ) = <. (/) , A >. )
10		opeq2 3805	. . . 4 |- ( x = B -> <. (/) , x >. = <. (/) , B >. )
11		elex 2771	. . . . 5 |- ( B e. W -> B e. _V )
12	11	adantl 277	. . . 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 4257	. . . . 5 |- ( ( (/) e. _V /\ B e. W ) -> <. (/) , B >. e. _V )
15	13, 14	sylancom 420	. . . 4 |- ( ( A e. V /\ B e. W ) -> <. (/) , B >. e. _V )
16	1, 10, 12, 15	fvmptd3 5651	. . 3 |- ( ( A e. V /\ B e. W ) -> (inl ` B ) = <. (/) , B >. )
17	9, 16	eqeq12d 2208	. 2 |- ( ( A e. V /\ B e. W ) -> ( (inl ` A ) = (inl ` B ) <-> <. (/) , A >. = <. (/) , B >. ) )
18		opthg 4267	. . . . 5 |- ( ( (/) e. _V /\ A e. V ) -> ( <. (/) , A >. = <. (/) , B >. <-> ( (/) = (/) /\ A = B ) ) )
19	5, 18	mpan 424	. . . 4 |- ( A e. V -> ( <. (/) , A >. = <. (/) , B >. <-> ( (/) = (/) /\ A = B ) ) )
20		eqid 2193	. . . . 5 |- (/) = (/)
21	20	biantrur 303	. . . 4 |- ( A = B <-> ( (/) = (/) /\ A = B ) )
22	19, 21	bitr4di 198	. . 3 |- ( A e. V -> ( <. (/) , A >. = <. (/) , B >. <-> A = B ) )
23	22	adantr 276	. 2 |- ( ( A e. V /\ B e. W ) -> ( <. (/) , A >. = <. (/) , B >. <-> A = B ) )
24	17, 23	bitrd 188	1 |- ( ( A e. V /\ B e. W ) -> ( (inl ` A ) = (inl ` B ) <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   _Vcvv 2760   (/)c0 3446   <.cop 3621   ` cfv 5254  inlcinl 7104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-inl 7106

This theorem is referenced by:  omp1eomlem  7153  difinfsnlem  7158