Theorem fprb 31669

Description: A condition for functionhood over a pair. (Contributed by Scott Fenton, 16-Sep-2013.)

Hypotheses

Ref	Expression
fprb.1	|- A e. _V
fprb.2	|- B e. _V

Assertion

Ref	Expression
fprb	|- ( A =/= B -> ( F : { A , B } --> R <-> E. x e. R E. y e. R F = { <. A , x >. , <. B , y >. } ) )

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

Proof of Theorem fprb

Step	Hyp	Ref	Expression
1		fprb.1	. . . . . . 7 |- A e. _V
2	1	prid1 4297	. . . . . 6 |- A e. { A , B }
3		ffvelrn 6357	. . . . . 6 |- ( ( F : { A , B } --> R /\ A e. { A , B } ) -> ( F ` A ) e. R )
4	2, 3	mpan2 707	. . . . 5 |- ( F : { A , B } --> R -> ( F ` A ) e. R )
5	4	adantr 481	. . . 4 |- ( ( F : { A , B } --> R /\ A =/= B ) -> ( F ` A ) e. R )
6		fprb.2	. . . . . . 7 |- B e. _V
7	6	prid2 4298	. . . . . 6 |- B e. { A , B }
8		ffvelrn 6357	. . . . . 6 |- ( ( F : { A , B } --> R /\ B e. { A , B } ) -> ( F ` B ) e. R )
9	7, 8	mpan2 707	. . . . 5 |- ( F : { A , B } --> R -> ( F ` B ) e. R )
10	9	adantr 481	. . . 4 |- ( ( F : { A , B } --> R /\ A =/= B ) -> ( F ` B ) e. R )
11		fvex 6201	. . . . . . . 8 |- ( F ` A ) e. _V
12	1, 11	fvpr1 6456	. . . . . . 7 |- ( A =/= B -> ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` A ) = ( F ` A ) )
13		fvex 6201	. . . . . . . 8 |- ( F ` B ) e. _V
14	6, 13	fvpr2 6457	. . . . . . 7 |- ( A =/= B -> ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` B ) = ( F ` B ) )
15		fveq2 6191	. . . . . . . . . 10 |- ( x = A -> ( F ` x ) = ( F ` A ) )
16		fveq2 6191	. . . . . . . . . 10 |- ( x = A -> ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` A ) )
17	15, 16	eqeq12d 2637	. . . . . . . . 9 |- ( x = A -> ( ( F ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) <-> ( F ` A ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` A ) ) )
18		eqcom 2629	. . . . . . . . 9 |- ( ( F ` A ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` A ) <-> ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` A ) = ( F ` A ) )
19	17, 18	syl6bb 276	. . . . . . . 8 |- ( x = A -> ( ( F ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) <-> ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` A ) = ( F ` A ) ) )
20		fveq2 6191	. . . . . . . . . 10 |- ( x = B -> ( F ` x ) = ( F ` B ) )
21		fveq2 6191	. . . . . . . . . 10 |- ( x = B -> ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` B ) )
22	20, 21	eqeq12d 2637	. . . . . . . . 9 |- ( x = B -> ( ( F ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) <-> ( F ` B ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` B ) ) )
23		eqcom 2629	. . . . . . . . 9 |- ( ( F ` B ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` B ) <-> ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` B ) = ( F ` B ) )
24	22, 23	syl6bb 276	. . . . . . . 8 |- ( x = B -> ( ( F ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) <-> ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` B ) = ( F ` B ) ) )
25	1, 6, 19, 24	ralpr 4238	. . . . . . 7 |- ( A. x e. { A , B } ( F ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) <-> ( ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` A ) = ( F ` A ) /\ ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` B ) = ( F ` B ) ) )
26	12, 14, 25	sylanbrc 698	. . . . . 6 |- ( A =/= B -> A. x e. { A , B } ( F ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) )
27	26	adantl 482	. . . . 5 |- ( ( F : { A , B } --> R /\ A =/= B ) -> A. x e. { A , B } ( F ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) )
28		ffn 6045	. . . . . 6 |- ( F : { A , B } --> R -> F Fn { A , B } )
29	1, 6, 11, 13	fpr 6421	. . . . . . 7 |- ( A =/= B -> { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } : { A , B } --> { ( F ` A ) , ( F ` B ) } )
30		ffn 6045	. . . . . . 7 |- ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } : { A , B } --> { ( F ` A ) , ( F ` B ) } -> { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } Fn { A , B } )
31	29, 30	syl 17	. . . . . 6 |- ( A =/= B -> { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } Fn { A , B } )
32		eqfnfv 6311	. . . . . 6 |- ( ( F Fn { A , B } /\ { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } Fn { A , B } ) -> ( F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } <-> A. x e. { A , B } ( F ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) ) )
33	28, 31, 32	syl2an 494	. . . . 5 |- ( ( F : { A , B } --> R /\ A =/= B ) -> ( F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } <-> A. x e. { A , B } ( F ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) ) )
34	27, 33	mpbird 247	. . . 4 |- ( ( F : { A , B } --> R /\ A =/= B ) -> F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } )
35		opeq2 4403	. . . . . . 7 |- ( x = ( F ` A ) -> <. A , x >. = <. A , ( F ` A ) >. )
36	35	preq1d 4274	. . . . . 6 |- ( x = ( F ` A ) -> { <. A , x >. , <. B , y >. } = { <. A , ( F ` A ) >. , <. B , y >. } )
37	36	eqeq2d 2632	. . . . 5 |- ( x = ( F ` A ) -> ( F = { <. A , x >. , <. B , y >. } <-> F = { <. A , ( F ` A ) >. , <. B , y >. } ) )
38		opeq2 4403	. . . . . . 7 |- ( y = ( F ` B ) -> <. B , y >. = <. B , ( F ` B ) >. )
39	38	preq2d 4275	. . . . . 6 |- ( y = ( F ` B ) -> { <. A , ( F ` A ) >. , <. B , y >. } = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } )
40	39	eqeq2d 2632	. . . . 5 |- ( y = ( F ` B ) -> ( F = { <. A , ( F ` A ) >. , <. B , y >. } <-> F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) )
41	37, 40	rspc2ev 3324	. . . 4 |- ( ( ( F ` A ) e. R /\ ( F ` B ) e. R /\ F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) -> E. x e. R E. y e. R F = { <. A , x >. , <. B , y >. } )
42	5, 10, 34, 41	syl3anc 1326	. . 3 |- ( ( F : { A , B } --> R /\ A =/= B ) -> E. x e. R E. y e. R F = { <. A , x >. , <. B , y >. } )
43	42	expcom 451	. 2 |- ( A =/= B -> ( F : { A , B } --> R -> E. x e. R E. y e. R F = { <. A , x >. , <. B , y >. } ) )
44		vex 3203	. . . . . . 7 |- x e. _V
45		vex 3203	. . . . . . 7 |- y e. _V
46	1, 6, 44, 45	fpr 6421	. . . . . 6 |- ( A =/= B -> { <. A , x >. , <. B , y >. } : { A , B } --> { x , y } )
47		prssi 4353	. . . . . 6 |- ( ( x e. R /\ y e. R ) -> { x , y } C_ R )
48		fss 6056	. . . . . 6 |- ( ( { <. A , x >. , <. B , y >. } : { A , B } --> { x , y } /\ { x , y } C_ R ) -> { <. A , x >. , <. B , y >. } : { A , B } --> R )
49	46, 47, 48	syl2an 494	. . . . 5 |- ( ( A =/= B /\ ( x e. R /\ y e. R ) ) -> { <. A , x >. , <. B , y >. } : { A , B } --> R )
50	49	ex 450	. . . 4 |- ( A =/= B -> ( ( x e. R /\ y e. R ) -> { <. A , x >. , <. B , y >. } : { A , B } --> R ) )
51		feq1 6026	. . . . 5 |- ( F = { <. A , x >. , <. B , y >. } -> ( F : { A , B } --> R <-> { <. A , x >. , <. B , y >. } : { A , B } --> R ) )
52	51	biimprcd 240	. . . 4 |- ( { <. A , x >. , <. B , y >. } : { A , B } --> R -> ( F = { <. A , x >. , <. B , y >. } -> F : { A , B } --> R ) )
53	50, 52	syl6 35	. . 3 |- ( A =/= B -> ( ( x e. R /\ y e. R ) -> ( F = { <. A , x >. , <. B , y >. } -> F : { A , B } --> R ) ) )
54	53	rexlimdvv 3037	. 2 |- ( A =/= B -> ( E. x e. R E. y e. R F = { <. A , x >. , <. B , y >. } -> F : { A , B } --> R ) )
55	43, 54	impbid 202	1 |- ( A =/= B -> ( F : { A , B } --> R <-> E. x e. R E. y e. R F = { <. A , x >. , <. B , y >. } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   {cpr 4179   <.cop 4183   Fn wfn 5883   -->wf 5884   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896

This theorem is referenced by: (None)