Theorem f1ompt 5663

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 5361	. . . . 5 |- ( F : A --> B -> F Fn A )
2		dff1o4 5465	. . . . . 6 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) )
3	2	baib 919	. . . . 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 5328	. . . . . 6 |- ( ( `' F |` B ) Fn B <-> A. y e. B E! z y `' F z )
6		nfcv 2319	. . . . . . . . . 10 |- F/_ x z
7		fmpt.1	. . . . . . . . . . 11 |- F = ( x e. A |-> C )
8		nfmpt1 4093	. . . . . . . . . . 11 |- F/_ x ( x e. A |-> C )
9	7, 8	nfcxfr 2316	. . . . . . . . . 10 |- F/_ x F
10		nfcv 2319	. . . . . . . . . 10 |- F/_ x y
11	6, 9, 10	nfbr 4046	. . . . . . . . 9 |- F/ x z F y
12		nfv 1528	. . . . . . . . 9 |- F/ z ( x e. A /\ y = C )
13		breq1 4003	. . . . . . . . . 10 |- ( z = x -> ( z F y <-> x F y ) )
14		df-mpt 4063	. . . . . . . . . . . . 13 |- ( x e. A |-> C ) = { <. x , y >. | ( x e. A /\ y = C ) }
15	7, 14	eqtri 2198	. . . . . . . . . . . 12 |- F = { <. x , y >. | ( x e. A /\ y = C ) }
16	15	breqi 4006	. . . . . . . . . . 11 |- ( x F y <-> x { <. x , y >. | ( x e. A /\ y = C ) } y )
17		df-br 4001	. . . . . . . . . . . 12 |- ( x { <. x , y >. | ( x e. A /\ y = C ) } y <-> <. x , y >. e. { <. x , y >. | ( x e. A /\ y = C ) } )
18		opabid 4254	. . . . . . . . . . . 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 2050	. . . . . . . 8 |- ( E! z z F y <-> E! x ( x e. A /\ y = C ) )
23		vex 2740	. . . . . . . . . 10 |- y e. _V
24		vex 2740	. . . . . . . . . 10 |- z e. _V
25	23, 24	brcnv 4806	. . . . . . . . 9 |- ( y `' F z <-> z F y )
26	25	eubii 2035	. . . . . . . 8 |- ( E! z y `' F z <-> E! z z F y )
27		df-reu 2462	. . . . . . . 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 2483	. . . . . 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 5002	. . . . . . 7 |- Rel `' F
32		df-rn 4634	. . . . . . . 8 |- ran F = dom `' F
33		frn 5370	. . . . . . . 8 |- ( F : A --> B -> ran F C_ B )
34	32, 33	eqsstrrid 3202	. . . . . . 7 |- ( F : A --> B -> dom `' F C_ B )
35		relssres 4941	. . . . . . 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 5302	. . . . 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 5457	. . 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 5662	. . 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 1353   E!weu 2026   e. wcel 2148   A.wral 2455   E!wreu 2457   C_ wss 3129   <.cop 3594   class class class wbr 4000   {copab 4060   |-> cmpt 4061   `'ccnv 4622   dom cdm 4623   ran crn 4624   |` cres 4625   Rel wrel 4628   Fn wfn 5207   -->wf 5208   -1-1-onto->wf1o 5211

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220

This theorem is referenced by:  xpf1o  6838  icoshftf1o  9978