Theorem fnprb 6131

Description: A function whose domain has at most two elements can be represented as a set of at most two ordered pairs. (Contributed by FL, 26-Jun-2011.) (Proof shortened by Scott Fenton, 12-Oct-2017.) Revised to eliminate unnecessary antecedent A =/= B. (Revised by NM, 29-Dec-2018.)

Hypotheses

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

Assertion

Ref	Expression
fnprb	|- ( F Fn { A , B } <-> F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } )

Proof of Theorem fnprb

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnprb.1	. . . . . 6 |- A e. _V
2	1	fnsnb 6091	. . . . 5 |- ( F Fn { A } <-> F = { <. A , ( F ` A ) >. } )
3		dfsn2 4045	. . . . . 6 |- { A } = { A , A }
4	3	fneq2i 5682	. . . . 5 |- ( F Fn { A } <-> F Fn { A , A } )
5		dfsn2 4045	. . . . . 6 |- { <. A , ( F ` A ) >. } = { <. A , ( F ` A ) >. , <. A , ( F ` A ) >. }
6	5	eqeq2i 2475	. . . . 5 |- ( F = { <. A , ( F ` A ) >. } <-> F = { <. A , ( F ` A ) >. , <. A , ( F ` A ) >. } )
7	2, 4, 6	3bitr3i 275	. . . 4 |- ( F Fn { A , A } <-> F = { <. A , ( F ` A ) >. , <. A , ( F ` A ) >. } )
8	7	a1i 11	. . 3 |- ( A = B -> ( F Fn { A , A } <-> F = { <. A , ( F ` A ) >. , <. A , ( F ` A ) >. } ) )
9		preq2 4112	. . . 4 |- ( A = B -> { A , A } = { A , B } )
10	9	fneq2d 5678	. . 3 |- ( A = B -> ( F Fn { A , A } <-> F Fn { A , B } ) )
11		id 22	. . . . . 6 |- ( A = B -> A = B )
12		fveq2 5872	. . . . . 6 |- ( A = B -> ( F ` A ) = ( F ` B ) )
13	11, 12	opeq12d 4227	. . . . 5 |- ( A = B -> <. A , ( F ` A ) >. = <. B , ( F ` B ) >. )
14	13	preq2d 4118	. . . 4 |- ( A = B -> { <. A , ( F ` A ) >. , <. A , ( F ` A ) >. } = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } )
15	14	eqeq2d 2471	. . 3 |- ( A = B -> ( F = { <. A , ( F ` A ) >. , <. A , ( F ` A ) >. } <-> F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) )
16	8, 10, 15	3bitr3d 283	. 2 |- ( A = B -> ( F Fn { A , B } <-> F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) )
17		fndm 5686	. . . . . 6 |- ( F Fn { A , B } -> dom F = { A , B } )
18		fvex 5882	. . . . . . 7 |- ( F ` A ) e. _V
19		fvex 5882	. . . . . . 7 |- ( F ` B ) e. _V
20	18, 19	dmprop 5489	. . . . . 6 |- dom { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } = { A , B }
21	17, 20	syl6eqr 2516	. . . . 5 |- ( F Fn { A , B } -> dom F = dom { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } )
22	21	adantl 466	. . . 4 |- ( ( A =/= B /\ F Fn { A , B } ) -> dom F = dom { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } )
23	17	adantl 466	. . . . . . 7 |- ( ( A =/= B /\ F Fn { A , B } ) -> dom F = { A , B } )
24	23	eleq2d 2527	. . . . . 6 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( x e. dom F <-> x e. { A , B } ) )
25		vex 3112	. . . . . . . 8 |- x e. _V
26	25	elpr 4050	. . . . . . 7 |- ( x e. { A , B } <-> ( x = A \/ x = B ) )
27	1, 18	fvpr1 6115	. . . . . . . . . . 11 |- ( A =/= B -> ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` A ) = ( F ` A ) )
28	27	adantr 465	. . . . . . . . . 10 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` A ) = ( F ` A ) )
29	28	eqcomd 2465	. . . . . . . . 9 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( F ` A ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` A ) )
30		fveq2 5872	. . . . . . . . . 10 |- ( x = A -> ( F ` x ) = ( F ` A ) )
31		fveq2 5872	. . . . . . . . . 10 |- ( x = A -> ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` A ) )
32	30, 31	eqeq12d 2479	. . . . . . . . 9 |- ( x = A -> ( ( F ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) <-> ( F ` A ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` A ) ) )
33	29, 32	syl5ibrcom 222	. . . . . . . 8 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( x = A -> ( F ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) ) )
34		fnprb.2	. . . . . . . . . . . 12 |- B e. _V
35	34, 19	fvpr2 6116	. . . . . . . . . . 11 |- ( A =/= B -> ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` B ) = ( F ` B ) )
36	35	adantr 465	. . . . . . . . . 10 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` B ) = ( F ` B ) )
37	36	eqcomd 2465	. . . . . . . . 9 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( F ` B ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` B ) )
38		fveq2 5872	. . . . . . . . . 10 |- ( x = B -> ( F ` x ) = ( F ` B ) )
39		fveq2 5872	. . . . . . . . . 10 |- ( x = B -> ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` B ) )
40	38, 39	eqeq12d 2479	. . . . . . . . 9 |- ( x = B -> ( ( F ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) <-> ( F ` B ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` B ) ) )
41	37, 40	syl5ibrcom 222	. . . . . . . 8 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( x = B -> ( F ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) ) )
42	33, 41	jaod 380	. . . . . . 7 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( ( x = A \/ x = B ) -> ( F ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) ) )
43	26, 42	syl5bi 217	. . . . . 6 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( x e. { A , B } -> ( F ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) ) )
44	24, 43	sylbid 215	. . . . 5 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( x e. dom F -> ( F ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) ) )
45	44	ralrimiv 2869	. . . 4 |- ( ( A =/= B /\ F Fn { A , B } ) -> A. x e. dom F ( F ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) )
46		fnfun 5684	. . . . 5 |- ( F Fn { A , B } -> Fun F )
47	1, 34, 18, 19	funpr 5645	. . . . 5 |- ( A =/= B -> Fun { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } )
48		eqfunfv 5987	. . . . 5 |- ( ( Fun F /\ Fun { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) -> ( F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } <-> ( dom F = dom { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } /\ A. x e. dom F ( F ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) ) ) )
49	46, 47, 48	syl2anr 478	. . . 4 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } <-> ( dom F = dom { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } /\ A. x e. dom F ( F ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) ) ) )
50	22, 45, 49	mpbir2and 922	. . 3 |- ( ( A =/= B /\ F Fn { A , B } ) -> F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } )
51	20	a1i 11	. . . . 5 |- ( A =/= B -> dom { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } = { A , B } )
52		df-fn 5597	. . . . 5 |- ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } Fn { A , B } <-> ( Fun { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } /\ dom { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } = { A , B } ) )
53	47, 51, 52	sylanbrc 664	. . . 4 |- ( A =/= B -> { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } Fn { A , B } )
54		fneq1 5675	. . . . 5 |- ( F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } -> ( F Fn { A , B } <-> { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } Fn { A , B } ) )
55	54	biimprd 223	. . . 4 |- ( F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } -> ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } Fn { A , B } -> F Fn { A , B } ) )
56	53, 55	mpan9 469	. . 3 |- ( ( A =/= B /\ F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) -> F Fn { A , B } )
57	50, 56	impbida 832	. 2 |- ( A =/= B -> ( F Fn { A , B } <-> F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) )
58	16, 57	pm2.61ine 2770	1 |- ( F Fn { A , B } <-> F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } )

Syntax hints:   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1395   e. wcel 1819   =/= wne 2652   A.wral 2807   _Vcvv 3109   {csn 4032   {cpr 4034   <.cop 4038   dom cdm 5008   Fun wfun 5588   Fn wfn 5589   ` cfv 5594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602

This theorem is referenced by:  wrd2pr2op  12897