Theorem inl11 7042

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 7024	. . . 4 |- inl = ( x e. _V |-> <. (/) , x >. )
2		opeq2 3766	. . . 4 |- ( x = A -> <. (/) , x >. = <. (/) , A >. )
3		elex 2741	. . . . 5 |- ( A e. V -> A e. _V )
4	3	adantr 274	. . . 4 |- ( ( A e. V /\ B e. W ) -> A e. _V )
5		0ex 4116	. . . . 5 |- (/) e. _V
6		simpl 108	. . . . 5 |- ( ( A e. V /\ B e. W ) -> A e. V )
7		opexg 4213	. . . . 5 |- ( ( (/) e. _V /\ A e. V ) -> <. (/) , A >. e. _V )
8	5, 6, 7	sylancr 412	. . . 4 |- ( ( A e. V /\ B e. W ) -> <. (/) , A >. e. _V )
9	1, 2, 4, 8	fvmptd3 5589	. . 3 |- ( ( A e. V /\ B e. W ) -> (inl ` A ) = <. (/) , A >. )
10		opeq2 3766	. . . 4 |- ( x = B -> <. (/) , x >. = <. (/) , B >. )
11		elex 2741	. . . . 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 4213	. . . . 5 |- ( ( (/) e. _V /\ B e. W ) -> <. (/) , B >. e. _V )
15	13, 14	sylancom 418	. . . 4 |- ( ( A e. V /\ B e. W ) -> <. (/) , B >. e. _V )
16	1, 10, 12, 15	fvmptd3 5589	. . 3 |- ( ( A e. V /\ B e. W ) -> (inl ` B ) = <. (/) , B >. )
17	9, 16	eqeq12d 2185	. 2 |- ( ( A e. V /\ B e. W ) -> ( (inl ` A ) = (inl ` B ) <-> <. (/) , A >. = <. (/) , B >. ) )
18		opthg 4223	. . . . 5 |- ( ( (/) e. _V /\ A e. V ) -> ( <. (/) , A >. = <. (/) , B >. <-> ( (/) = (/) /\ A = B ) ) )
19	5, 18	mpan 422	. . . 4 |- ( A e. V -> ( <. (/) , A >. = <. (/) , B >. <-> ( (/) = (/) /\ A = B ) ) )
20		eqid 2170	. . . . 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 1348   e. wcel 2141   _Vcvv 2730   (/)c0 3414   <.cop 3586   ` cfv 5198  inlcinl 7022

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-inl 7024

This theorem is referenced by:  omp1eomlem  7071  difinfsnlem  7076