Theorem f1omptsnlem 33183

Description: This is the core of the proof of f1omptsn 33184, but to avoid the distinct variables on the definitions, we split this proof into two. (Contributed by ML, 15-Jul-2020.)

Hypotheses

Ref	Expression
f1omptsn.f	|- F = ( x e. A |-> { x } )
f1omptsn.r	|- R = { u | E. x e. A u = { x } }

Assertion

Ref	Expression
f1omptsnlem	|- F : A -1-1-onto-> R

Distinct variable groups:   x, A, u   x, F   u, R, x

Allowed substitution hint:   F( u)

Proof of Theorem f1omptsnlem

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1omptsn.f	. . . . 5 |- F = ( x e. A |-> { x } )
2		eqid 2622	. . . . . . 7 |- { x } = { x }
3		snex 4908	. . . . . . . 8 |- { x } e. _V
4		eqsbc3 3475	. . . . . . . 8 |- ( { x } e. _V -> ( [. { x } / u ]. u = { x } <-> { x } = { x } ) )
5	3, 4	ax-mp 5	. . . . . . 7 |- ( [. { x } / u ]. u = { x } <-> { x } = { x } )
6	2, 5	mpbir 221	. . . . . 6 |- [. { x } / u ]. u = { x }
7		sbcel2 3989	. . . . . . . 8 |- ( [. { x } / u ]. x e. A <-> x e. [_ { x } / u ]_ A )
8		csbconstg 3546	. . . . . . . . . 10 |- ( { x } e. _V -> [_ { x } / u ]_ A = A )
9	3, 8	ax-mp 5	. . . . . . . . 9 |- [_ { x } / u ]_ A = A
10	9	eleq2i 2693	. . . . . . . 8 |- ( x e. [_ { x } / u ]_ A <-> x e. A )
11	7, 10	bitri 264	. . . . . . 7 |- ( [. { x } / u ]. x e. A <-> x e. A )
12		f1omptsn.r	. . . . . . . . . . . . . 14 |- R = { u | E. x e. A u = { x } }
13	12	abeq2i 2735	. . . . . . . . . . . . 13 |- ( u e. R <-> E. x e. A u = { x } )
14		df-rex 2918	. . . . . . . . . . . . 13 |- ( E. x e. A u = { x } <-> E. x ( x e. A /\ u = { x } ) )
15	13, 14	sylbbr 226	. . . . . . . . . . . 12 |- ( E. x ( x e. A /\ u = { x } ) -> u e. R )
16	15	19.23bi 2061	. . . . . . . . . . 11 |- ( ( x e. A /\ u = { x } ) -> u e. R )
17	16	sbcth 3450	. . . . . . . . . 10 |- ( { x } e. _V -> [. { x } / u ]. ( ( x e. A /\ u = { x } ) -> u e. R ) )
18	3, 17	ax-mp 5	. . . . . . . . 9 |- [. { x } / u ]. ( ( x e. A /\ u = { x } ) -> u e. R )
19		sbcimg 3477	. . . . . . . . . 10 |- ( { x } e. _V -> ( [. { x } / u ]. ( ( x e. A /\ u = { x } ) -> u e. R ) <-> ( [. { x } / u ]. ( x e. A /\ u = { x } ) -> [. { x } / u ]. u e. R ) ) )
20	3, 19	ax-mp 5	. . . . . . . . 9 |- ( [. { x } / u ]. ( ( x e. A /\ u = { x } ) -> u e. R ) <-> ( [. { x } / u ]. ( x e. A /\ u = { x } ) -> [. { x } / u ]. u e. R ) )
21	18, 20	mpbi 220	. . . . . . . 8 |- ( [. { x } / u ]. ( x e. A /\ u = { x } ) -> [. { x } / u ]. u e. R )
22		sbcan 3478	. . . . . . . 8 |- ( [. { x } / u ]. ( x e. A /\ u = { x } ) <-> ( [. { x } / u ]. x e. A /\ [. { x } / u ]. u = { x } ) )
23		sbcel1v 3495	. . . . . . . 8 |- ( [. { x } / u ]. u e. R <-> { x } e. R )
24	21, 22, 23	3imtr3i 280	. . . . . . 7 |- ( ( [. { x } / u ]. x e. A /\ [. { x } / u ]. u = { x } ) -> { x } e. R )
25	11, 24	sylanbr 490	. . . . . 6 |- ( ( x e. A /\ [. { x } / u ]. u = { x } ) -> { x } e. R )
26	6, 25	mpan2 707	. . . . 5 |- ( x e. A -> { x } e. R )
27	1, 26	fmpti 6383	. . . 4 |- F : A --> R
28	1	fvmpt2 6291	. . . . . . . . 9 |- ( ( x e. A /\ { x } e. R ) -> ( F ` x ) = { x } )
29	26, 28	mpdan 702	. . . . . . . 8 |- ( x e. A -> ( F ` x ) = { x } )
30		sneq 4187	. . . . . . . . 9 |- ( x = y -> { x } = { y } )
31	30, 1, 3	fvmpt3i 6287	. . . . . . . 8 |- ( y e. A -> ( F ` y ) = { y } )
32	29, 31	eqeqan12d 2638	. . . . . . 7 |- ( ( x e. A /\ y e. A ) -> ( ( F ` x ) = ( F ` y ) <-> { x } = { y } ) )
33		vex 3203	. . . . . . . 8 |- x e. _V
34		sneqbg 4374	. . . . . . . 8 |- ( x e. _V -> ( { x } = { y } <-> x = y ) )
35	33, 34	ax-mp 5	. . . . . . 7 |- ( { x } = { y } <-> x = y )
36	32, 35	syl6bb 276	. . . . . 6 |- ( ( x e. A /\ y e. A ) -> ( ( F ` x ) = ( F ` y ) <-> x = y ) )
37	36	biimpd 219	. . . . 5 |- ( ( x e. A /\ y e. A ) -> ( ( F ` x ) = ( F ` y ) -> x = y ) )
38	37	rgen2a 2977	. . . 4 |- A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y )
39		dff13 6512	. . . 4 |- ( F : A -1-1-> R <-> ( F : A --> R /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) )
40	27, 38, 39	mpbir2an 955	. . 3 |- F : A -1-1-> R
41		f1f1orn 6148	. . 3 |- ( F : A -1-1-> R -> F : A -1-1-onto-> ran F )
42	40, 41	ax-mp 5	. 2 |- F : A -1-1-onto-> ran F
43		rnmptsn 33182	. . . 4 |- ran ( x e. A |-> { x } ) = { u | E. x e. A u = { x } }
44	1	rneqi 5352	. . . 4 |- ran F = ran ( x e. A |-> { x } )
45	43, 44, 12	3eqtr4i 2654	. . 3 |- ran F = R
46		f1oeq3 6129	. . 3 |- ( ran F = R -> ( F : A -1-1-onto-> ran F <-> F : A -1-1-onto-> R ) )
47	45, 46	ax-mp 5	. 2 |- ( F : A -1-1-onto-> ran F <-> F : A -1-1-onto-> R )
48	42, 47	mpbi 220	1 |- F : A -1-1-onto-> R

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   _Vcvv 3200   [.wsbc 3435   [_csb 3533   {csn 4177   |-> cmpt 4729   ran crn 5115   -->wf 5884   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by:  f1omptsn  33184