Theorem f1omptsn 33184

Description: A function mapping to singletons is bijective onto a set of singletons. (Contributed by ML, 16-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
f1omptsn	|- F : A -1-1-onto-> R

Distinct variable group:   u, A, x

Allowed substitution hints:   R( x, u)   F( x, u)

Proof of Theorem f1omptsn

Dummy variables a z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 4187	. . . . . 6 |- ( x = a -> { x } = { a } )
2	1	cbvmptv 4750	. . . . 5 |- ( x e. A |-> { x } ) = ( a e. A |-> { a } )
3	2	eqcomi 2631	. . . 4 |- ( a e. A |-> { a } ) = ( x e. A |-> { x } )
4		id 22	. . . . . . . 8 |- ( u = z -> u = z )
5	4, 1	eqeqan12d 2638	. . . . . . 7 |- ( ( u = z /\ x = a ) -> ( u = { x } <-> z = { a } ) )
6	5	cbvrexdva 3178	. . . . . 6 |- ( u = z -> ( E. x e. A u = { x } <-> E. a e. A z = { a } ) )
7	6	cbvabv 2747	. . . . 5 |- { u | E. x e. A u = { x } } = { z | E. a e. A z = { a } }
8	7	eqcomi 2631	. . . 4 |- { z | E. a e. A z = { a } } = { u | E. x e. A u = { x } }
9	3, 8	f1omptsnlem 33183	. . 3 |- ( a e. A |-> { a } ) : A -1-1-onto-> { z | E. a e. A z = { a } }
10		f1omptsn.r	. . . . 5 |- R = { u | E. x e. A u = { x } }
11	10, 7	eqtri 2644	. . . 4 |- R = { z | E. a e. A z = { a } }
12		f1oeq3 6129	. . . 4 |- ( R = { z | E. a e. A z = { a } } -> ( ( a e. A |-> { a } ) : A -1-1-onto-> R <-> ( a e. A |-> { a } ) : A -1-1-onto-> { z | E. a e. A z = { a } } ) )
13	11, 12	ax-mp 5	. . 3 |- ( ( a e. A |-> { a } ) : A -1-1-onto-> R <-> ( a e. A |-> { a } ) : A -1-1-onto-> { z | E. a e. A z = { a } } )
14	9, 13	mpbir 221	. 2 |- ( a e. A |-> { a } ) : A -1-1-onto-> R
15		f1omptsn.f	. . . 4 |- F = ( x e. A |-> { x } )
16	15, 2	eqtri 2644	. . 3 |- F = ( a e. A |-> { a } )
17		f1oeq1 6127	. . 3 |- ( F = ( a e. A |-> { a } ) -> ( F : A -1-1-onto-> R <-> ( a e. A |-> { a } ) : A -1-1-onto-> R ) )
18	16, 17	ax-mp 5	. 2 |- ( F : A -1-1-onto-> R <-> ( a e. A |-> { a } ) : A -1-1-onto-> R )
19	14, 18	mpbir 221	1 |- F : A -1-1-onto-> R

Syntax hints:   <-> wb 196   = wceq 1483   {cab 2608   E.wrex 2913   {csn 4177   |-> cmpt 4729   -1-1-onto->wf1o 5887

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: (None)