Theorem f1ompt 5320

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 5046	. . . . 5 |- ( F : A --> B -> F Fn A )
2		dff1o4 5134	. . . . . 6 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) )
3	2	baib 828	. . . . 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 5015	. . . . . 6 |- ( ( `' F |` B ) Fn B <-> A. y e. B E! z y `' F z )
6		nfcv 2178	. . . . . . . . . 10 |- F/_ x z
7		fmpt.1	. . . . . . . . . . 11 |- F = ( x e. A |-> C )
8		nfmpt1 3850	. . . . . . . . . . 11 |- F/_ x ( x e. A |-> C )
9	7, 8	nfcxfr 2175	. . . . . . . . . 10 |- F/_ x F
10		nfcv 2178	. . . . . . . . . 10 |- F/_ x y
11	6, 9, 10	nfbr 3808	. . . . . . . . 9 |- F/ x z F y
12		nfv 1421	. . . . . . . . 9 |- F/ z ( x e. A /\ y = C )
13		breq1 3767	. . . . . . . . . 10 |- ( z = x -> ( z F y <-> x F y ) )
14		df-mpt 3820	. . . . . . . . . . . . 13 |- ( x e. A |-> C ) = { <. x , y >. | ( x e. A /\ y = C ) }
15	7, 14	eqtri 2060	. . . . . . . . . . . 12 |- F = { <. x , y >. | ( x e. A /\ y = C ) }
16	15	breqi 3770	. . . . . . . . . . 11 |- ( x F y <-> x { <. x , y >. | ( x e. A /\ y = C ) } y )
17		df-br 3765	. . . . . . . . . . . 12 |- ( x { <. x , y >. | ( x e. A /\ y = C ) } y <-> <. x , y >. e. { <. x , y >. | ( x e. A /\ y = C ) } )
18		opabid 3994	. . . . . . . . . . . 12 |- ( <. x , y >. e. { <. x , y >. | ( x e. A /\ y = C ) } <-> ( x e. A /\ y = C ) )
19	17, 18	bitri 173	. . . . . . . . . . 11 |- ( x { <. x , y >. | ( x e. A /\ y = C ) } y <-> ( x e. A /\ y = C ) )
20	16, 19	bitri 173	. . . . . . . . . 10 |- ( x F y <-> ( x e. A /\ y = C ) )
21	13, 20	syl6bb 185	. . . . . . . . 9 |- ( z = x -> ( z F y <-> ( x e. A /\ y = C ) ) )
22	11, 12, 21	cbveu 1924	. . . . . . . 8 |- ( E! z z F y <-> E! x ( x e. A /\ y = C ) )
23		vex 2560	. . . . . . . . . 10 |- y e. _V
24		vex 2560	. . . . . . . . . 10 |- z e. _V
25	23, 24	brcnv 4518	. . . . . . . . 9 |- ( y `' F z <-> z F y )
26	25	eubii 1909	. . . . . . . 8 |- ( E! z y `' F z <-> E! z z F y )
27		df-reu 2313	. . . . . . . 8 |- ( E! x e. A y = C <-> E! x ( x e. A /\ y = C ) )
28	22, 26, 27	3bitr4i 201	. . . . . . 7 |- ( E! z y `' F z <-> E! x e. A y = C )
29	28	ralbii 2330	. . . . . 6 |- ( A. y e. B E! z y `' F z <-> A. y e. B E! x e. A y = C )
30	5, 29	bitri 173	. . . . 5 |- ( ( `' F |` B ) Fn B <-> A. y e. B E! x e. A y = C )
31		relcnv 4703	. . . . . . 7 |- Rel `' F
32		df-rn 4356	. . . . . . . 8 |- ran F = dom `' F
33		frn 5052	. . . . . . . 8 |- ( F : A --> B -> ran F C_ B )
34	32, 33	syl5eqssr 2990	. . . . . . 7 |- ( F : A --> B -> dom `' F C_ B )
35		relssres 4648	. . . . . . 7 |- ( ( Rel `' F /\ dom `' F C_ B ) -> ( `' F |` B ) = `' F )
36	31, 34, 35	sylancr 393	. . . . . 6 |- ( F : A --> B -> ( `' F |` B ) = `' F )
37	36	fneq1d 4989	. . . . 5 |- ( F : A --> B -> ( ( `' F |` B ) Fn B <-> `' F Fn B ) )
38	30, 37	syl5bbr 183	. . . 4 |- ( F : A --> B -> ( A. y e. B E! x e. A y = C <-> `' F Fn B ) )
39	4, 38	bitr4d 180	. . 3 |- ( F : A --> B -> ( F : A -1-1-onto-> B <-> A. y e. B E! x e. A y = C ) )
40	39	pm5.32i 427	. 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 5126	. . 3 |- ( F : A -1-1-onto-> B -> F : A --> B )
42	41	pm4.71ri 372	. 2 |- ( F : A -1-1-onto-> B <-> ( F : A --> B /\ F : A -1-1-onto-> B ) )
43	7	fmpt 5319	. . 3 |- ( A. x e. A C e. B <-> F : A --> B )
44	43	anbi1i 431	. 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 201	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 97   <-> wb 98   = wceq 1243   e. wcel 1393   E!weu 1900   A.wral 2306   E!wreu 2308   C_ wss 2917   <.cop 3378   class class class wbr 3764   {copab 3817   |-> cmpt 3818   `'ccnv 4344   dom cdm 4345   ran crn 4346   |` cres 4347   Rel wrel 4350   Fn wfn 4897   -->wf 4898   -1-1-onto->wf1o 4901

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910

This theorem is referenced by:  icoshftf1o  8859