Theorem f1ompt 5647

Description: Express bijection for a mapping operation. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by Mario Carneiro, 4-Dec-2016.)

Hypothesis

Ref	Expression
fmpt.1	|- F = ( x e. A |-> C )

Assertion

Ref	Expression
f1ompt	|- ( F : A -1-1-onto-> B <-> ( A. x e. A C e. B /\ A. y e. B E! x e. A y = C ) )

Distinct variable groups:   x, y, A   x, B, y   y, C   y, F

Allowed substitution hints:   C( x)   F( x)

Proof of Theorem f1ompt

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffn 5347	. . . . 5 |- ( F : A --> B -> F Fn A )
2		dff1o4 5450	. . . . . 6 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) )
3	2	baib 914	. . . . 5 |- ( F Fn A -> ( F : A -1-1-onto-> B <-> `' F Fn B ) )
4	1, 3	syl 14	. . . 4 |- ( F : A --> B -> ( F : A -1-1-onto-> B <-> `' F Fn B ) )
5		fnres 5314	. . . . . 6 |- ( ( `' F |` B ) Fn B <-> A. y e. B E! z y `' F z )
6		nfcv 2312	. . . . . . . . . 10 |- F/_ x z
7		fmpt.1	. . . . . . . . . . 11 |- F = ( x e. A |-> C )
8		nfmpt1 4082	. . . . . . . . . . 11 |- F/_ x ( x e. A |-> C )
9	7, 8	nfcxfr 2309	. . . . . . . . . 10 |- F/_ x F
10		nfcv 2312	. . . . . . . . . 10 |- F/_ x y
11	6, 9, 10	nfbr 4035	. . . . . . . . 9 |- F/ x z F y
12		nfv 1521	. . . . . . . . 9 |- F/ z ( x e. A /\ y = C )
13		breq1 3992	. . . . . . . . . 10 |- ( z = x -> ( z F y <-> x F y ) )
14		df-mpt 4052	. . . . . . . . . . . . 13 |- ( x e. A |-> C ) = { <. x , y >. | ( x e. A /\ y = C ) }
15	7, 14	eqtri 2191	. . . . . . . . . . . 12 |- F = { <. x , y >. | ( x e. A /\ y = C ) }
16	15	breqi 3995	. . . . . . . . . . 11 |- ( x F y <-> x { <. x , y >. | ( x e. A /\ y = C ) } y )
17		df-br 3990	. . . . . . . . . . . 12 |- ( x { <. x , y >. | ( x e. A /\ y = C ) } y <-> <. x , y >. e. { <. x , y >. | ( x e. A /\ y = C ) } )
18		opabid 4242	. . . . . . . . . . . 12 |- ( <. x , y >. e. { <. x , y >. | ( x e. A /\ y = C ) } <-> ( x e. A /\ y = C ) )
19	17, 18	bitri 183	. . . . . . . . . . 11 |- ( x { <. x , y >. | ( x e. A /\ y = C ) } y <-> ( x e. A /\ y = C ) )
20	16, 19	bitri 183	. . . . . . . . . 10 |- ( x F y <-> ( x e. A /\ y = C ) )
21	13, 20	bitrdi 195	. . . . . . . . 9 |- ( z = x -> ( z F y <-> ( x e. A /\ y = C ) ) )
22	11, 12, 21	cbveu 2043	. . . . . . . 8 |- ( E! z z F y <-> E! x ( x e. A /\ y = C ) )
23		vex 2733	. . . . . . . . . 10 |- y e. _V
24		vex 2733	. . . . . . . . . 10 |- z e. _V
25	23, 24	brcnv 4794	. . . . . . . . 9 |- ( y `' F z <-> z F y )
26	25	eubii 2028	. . . . . . . 8 |- ( E! z y `' F z <-> E! z z F y )
27		df-reu 2455	. . . . . . . 8 |- ( E! x e. A y = C <-> E! x ( x e. A /\ y = C ) )
28	22, 26, 27	3bitr4i 211	. . . . . . 7 |- ( E! z y `' F z <-> E! x e. A y = C )
29	28	ralbii 2476	. . . . . 6 |- ( A. y e. B E! z y `' F z <-> A. y e. B E! x e. A y = C )
30	5, 29	bitri 183	. . . . 5 |- ( ( `' F |` B ) Fn B <-> A. y e. B E! x e. A y = C )
31		relcnv 4989	. . . . . . 7 |- Rel `' F
32		df-rn 4622	. . . . . . . 8 |- ran F = dom `' F
33		frn 5356	. . . . . . . 8 |- ( F : A --> B -> ran F C_ B )
34	32, 33	eqsstrrid 3194	. . . . . . 7 |- ( F : A --> B -> dom `' F C_ B )
35		relssres 4929	. . . . . . 7 |- ( ( Rel `' F /\ dom `' F C_ B ) -> ( `' F |` B ) = `' F )
36	31, 34, 35	sylancr 412	. . . . . 6 |- ( F : A --> B -> ( `' F |` B ) = `' F )
37	36	fneq1d 5288	. . . . 5 |- ( F : A --> B -> ( ( `' F |` B ) Fn B <-> `' F Fn B ) )
38	30, 37	bitr3id 193	. . . 4 |- ( F : A --> B -> ( A. y e. B E! x e. A y = C <-> `' F Fn B ) )
39	4, 38	bitr4d 190	. . 3 |- ( F : A --> B -> ( F : A -1-1-onto-> B <-> A. y e. B E! x e. A y = C ) )
40	39	pm5.32i 451	. 2 |- ( ( F : A --> B /\ F : A -1-1-onto-> B ) <-> ( F : A --> B /\ A. y e. B E! x e. A y = C ) )
41		f1of 5442	. . 3 |- ( F : A -1-1-onto-> B -> F : A --> B )
42	41	pm4.71ri 390	. 2 |- ( F : A -1-1-onto-> B <-> ( F : A --> B /\ F : A -1-1-onto-> B ) )
43	7	fmpt 5646	. . 3 |- ( A. x e. A C e. B <-> F : A --> B )
44	43	anbi1i 455	. 2 |- ( ( A. x e. A C e. B /\ A. y e. B E! x e. A y = C ) <-> ( F : A --> B /\ A. y e. B E! x e. A y = C ) )
45	40, 42, 44	3bitr4i 211	1 |- ( F : A -1-1-onto-> B <-> ( A. x e. A C e. B /\ A. y e. B E! x e. A y = C ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1348   E!weu 2019   e. wcel 2141   A.wral 2448   E!wreu 2450   C_ wss 3121   <.cop 3586   class class class wbr 3989   {copab 4049   |-> cmpt 4050   `'ccnv 4610   dom cdm 4611   ran crn 4612   |` cres 4613   Rel wrel 4616   Fn wfn 5193   -->wf 5194   -1-1-onto->wf1o 5197

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-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-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-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  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-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206

This theorem is referenced by:  xpf1o  6822  icoshftf1o  9948