Theorem f1ompt 5579

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 5280	. . . . 5 |- ( F : A --> B -> F Fn A )
2		dff1o4 5383	. . . . . 6 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) )
3	2	baib 905	. . . . 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 5247	. . . . . 6 |- ( ( `' F |` B ) Fn B <-> A. y e. B E! z y `' F z )
6		nfcv 2282	. . . . . . . . . 10 |- F/_ x z
7		fmpt.1	. . . . . . . . . . 11 |- F = ( x e. A |-> C )
8		nfmpt1 4029	. . . . . . . . . . 11 |- F/_ x ( x e. A |-> C )
9	7, 8	nfcxfr 2279	. . . . . . . . . 10 |- F/_ x F
10		nfcv 2282	. . . . . . . . . 10 |- F/_ x y
11	6, 9, 10	nfbr 3982	. . . . . . . . 9 |- F/ x z F y
12		nfv 1509	. . . . . . . . 9 |- F/ z ( x e. A /\ y = C )
13		breq1 3940	. . . . . . . . . 10 |- ( z = x -> ( z F y <-> x F y ) )
14		df-mpt 3999	. . . . . . . . . . . . 13 |- ( x e. A |-> C ) = { <. x , y >. | ( x e. A /\ y = C ) }
15	7, 14	eqtri 2161	. . . . . . . . . . . 12 |- F = { <. x , y >. | ( x e. A /\ y = C ) }
16	15	breqi 3943	. . . . . . . . . . 11 |- ( x F y <-> x { <. x , y >. | ( x e. A /\ y = C ) } y )
17		df-br 3938	. . . . . . . . . . . 12 |- ( x { <. x , y >. | ( x e. A /\ y = C ) } y <-> <. x , y >. e. { <. x , y >. | ( x e. A /\ y = C ) } )
18		opabid 4187	. . . . . . . . . . . 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	syl6bb 195	. . . . . . . . 9 |- ( z = x -> ( z F y <-> ( x e. A /\ y = C ) ) )
22	11, 12, 21	cbveu 2024	. . . . . . . 8 |- ( E! z z F y <-> E! x ( x e. A /\ y = C ) )
23		vex 2692	. . . . . . . . . 10 |- y e. _V
24		vex 2692	. . . . . . . . . 10 |- z e. _V
25	23, 24	brcnv 4730	. . . . . . . . 9 |- ( y `' F z <-> z F y )
26	25	eubii 2009	. . . . . . . 8 |- ( E! z y `' F z <-> E! z z F y )
27		df-reu 2424	. . . . . . . 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 2444	. . . . . 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 4925	. . . . . . 7 |- Rel `' F
32		df-rn 4558	. . . . . . . 8 |- ran F = dom `' F
33		frn 5289	. . . . . . . 8 |- ( F : A --> B -> ran F C_ B )
34	32, 33	eqsstrrid 3149	. . . . . . 7 |- ( F : A --> B -> dom `' F C_ B )
35		relssres 4865	. . . . . . 7 |- ( ( Rel `' F /\ dom `' F C_ B ) -> ( `' F |` B ) = `' F )
36	31, 34, 35	sylancr 411	. . . . . 6 |- ( F : A --> B -> ( `' F |` B ) = `' F )
37	36	fneq1d 5221	. . . . 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 450	. 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 5375	. . 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 5578	. . 3 |- ( A. x e. A C e. B <-> F : A --> B )
44	43	anbi1i 454	. 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 1332   e. wcel 1481   E!weu 2000   A.wral 2417   E!wreu 2419   C_ wss 3076   <.cop 3535   class class class wbr 3937   {copab 3996   |-> cmpt 3997   `'ccnv 4546   dom cdm 4547   ran crn 4548   |` cres 4549   Rel wrel 4552   Fn wfn 5126   -->wf 5127   -1-1-onto->wf1o 5130

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139

This theorem is referenced by:  xpf1o  6746  icoshftf1o  9804