Theorem f1ompt 5833

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 5513	. . . . 5 |- ( F : A --> B -> F Fn A )
2		dff1o4 5627	. . . . . 6 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) )
3	2	baib 927	. . . . 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 5480	. . . . . 6 |- ( ( `' F |` B ) Fn B <-> A. y e. B E! z y `' F z )
6		nfcv 2386	. . . . . . . . . 10 |- F/_ x z
7		fmpt.1	. . . . . . . . . . 11 |- F = ( x e. A |-> C )
8		nfmpt1 4208	. . . . . . . . . . 11 |- F/_ x ( x e. A |-> C )
9	7, 8	nfcxfr 2383	. . . . . . . . . 10 |- F/_ x F
10		nfcv 2386	. . . . . . . . . 10 |- F/_ x y
11	6, 9, 10	nfbr 4161	. . . . . . . . 9 |- F/ x z F y
12		nfv 1577	. . . . . . . . 9 |- F/ z ( x e. A /\ y = C )
13		breq1 4117	. . . . . . . . . 10 |- ( z = x -> ( z F y <-> x F y ) )
14		df-mpt 4178	. . . . . . . . . . . . 13 |- ( x e. A |-> C ) = { <. x , y >. | ( x e. A /\ y = C ) }
15	7, 14	eqtri 2255	. . . . . . . . . . . 12 |- F = { <. x , y >. | ( x e. A /\ y = C ) }
16	15	breqi 4120	. . . . . . . . . . 11 |- ( x F y <-> x { <. x , y >. | ( x e. A /\ y = C ) } y )
17		df-br 4115	. . . . . . . . . . . 12 |- ( x { <. x , y >. | ( x e. A /\ y = C ) } y <-> <. x , y >. e. { <. x , y >. | ( x e. A /\ y = C ) } )
18		opabid 4379	. . . . . . . . . . . 12 |- ( <. x , y >. e. { <. x , y >. | ( x e. A /\ y = C ) } <-> ( x e. A /\ y = C ) )
19	17, 18	bitri 184	. . . . . . . . . . 11 |- ( x { <. x , y >. | ( x e. A /\ y = C ) } y <-> ( x e. A /\ y = C ) )
20	16, 19	bitri 184	. . . . . . . . . 10 |- ( x F y <-> ( x e. A /\ y = C ) )
21	13, 20	bitrdi 196	. . . . . . . . 9 |- ( z = x -> ( z F y <-> ( x e. A /\ y = C ) ) )
22	11, 12, 21	cbveu 2106	. . . . . . . 8 |- ( E! z z F y <-> E! x ( x e. A /\ y = C ) )
23		vex 2818	. . . . . . . . . 10 |- y e. _V
24		vex 2818	. . . . . . . . . 10 |- z e. _V
25	23, 24	brcnv 4943	. . . . . . . . 9 |- ( y `' F z <-> z F y )
26	25	eubii 2091	. . . . . . . 8 |- ( E! z y `' F z <-> E! z z F y )
27		df-reu 2529	. . . . . . . 8 |- ( E! x e. A y = C <-> E! x ( x e. A /\ y = C ) )
28	22, 26, 27	3bitr4i 212	. . . . . . 7 |- ( E! z y `' F z <-> E! x e. A y = C )
29	28	ralbii 2550	. . . . . 6 |- ( A. y e. B E! z y `' F z <-> A. y e. B E! x e. A y = C )
30	5, 29	bitri 184	. . . . 5 |- ( ( `' F |` B ) Fn B <-> A. y e. B E! x e. A y = C )
31		relcnv 5145	. . . . . . 7 |- Rel `' F
32		df-rn 4765	. . . . . . . 8 |- ran F = dom `' F
33		frn 5522	. . . . . . . 8 |- ( F : A --> B -> ran F C_ B )
34	32, 33	eqsstrrid 3289	. . . . . . 7 |- ( F : A --> B -> dom `' F C_ B )
35		relssres 5081	. . . . . . 7 |- ( ( Rel `' F /\ dom `' F C_ B ) -> ( `' F |` B ) = `' F )
36	31, 34, 35	sylancr 414	. . . . . 6 |- ( F : A --> B -> ( `' F |` B ) = `' F )
37	36	fneq1d 5451	. . . . 5 |- ( F : A --> B -> ( ( `' F |` B ) Fn B <-> `' F Fn B ) )
38	30, 37	bitr3id 194	. . . 4 |- ( F : A --> B -> ( A. y e. B E! x e. A y = C <-> `' F Fn B ) )
39	4, 38	bitr4d 191	. . 3 |- ( F : A --> B -> ( F : A -1-1-onto-> B <-> A. y e. B E! x e. A y = C ) )
40	39	pm5.32i 454	. 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 5619	. . 3 |- ( F : A -1-1-onto-> B -> F : A --> B )
42	41	pm4.71ri 392	. 2 |- ( F : A -1-1-onto-> B <-> ( F : A --> B /\ F : A -1-1-onto-> B ) )
43	7	fmpt 5832	. . 3 |- ( A. x e. A C e. B <-> F : A --> B )
44	43	anbi1i 458	. 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 212	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 104   <-> wb 105   = wceq 1398   E!weu 2082   e. wcel 2205   A.wral 2522   E!wreu 2524   C_ wss 3214   <.cop 3697   class class class wbr 4114   {copab 4175   |-> cmpt 4176   `'ccnv 4753   dom cdm 4754   ran crn 4755   |` cres 4756   Rel wrel 4759   Fn wfn 5352   -->wf 5353   -1-1-onto->wf1o 5356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365

This theorem is referenced by:  xpf1o  7110  icoshftf1o  10343