Theorem fnprb 5934

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 5896	. . . . 5 |- ( F Fn { A } <-> F = { <. A , ( F ` A ) >. } )
3		dfsn2 3888	. . . . . 6 |- { A } = { A , A }
4	3	fneq2i 5504	. . . . 5 |- ( F Fn { A } <-> F Fn { A , A } )
5		dfsn2 3888	. . . . . 6 |- { <. A , ( F ` A ) >. } = { <. A , ( F ` A ) >. , <. A , ( F ` A ) >. }
6	5	eqeq2i 2451	. . . . 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 3953	. . . 4 |- ( A = B -> { A , A } = { A , B } )
10	9	fneq2d 5500	. . 3 |- ( A = B -> ( F Fn { A , A } <-> F Fn { A , B } ) )
11		id 22	. . . . . 6 |- ( A = B -> A = B )
12		fveq2 5689	. . . . . 6 |- ( A = B -> ( F ` A ) = ( F ` B ) )
13	11, 12	opeq12d 4065	. . . . 5 |- ( A = B -> <. A , ( F ` A ) >. = <. B , ( F ` B ) >. )
14	13	preq2d 3959	. . . 4 |- ( A = B -> { <. A , ( F ` A ) >. , <. A , ( F ` A ) >. } = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } )
15	14	eqeq2d 2452	. . 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 5508	. . . . . 6 |- ( F Fn { A , B } -> dom F = { A , B } )
18		fvex 5699	. . . . . . 7 |- ( F ` A ) e. _V
19		fvex 5699	. . . . . . 7 |- ( F ` B ) e. _V
20	18, 19	dmprop 5312	. . . . . 6 |- dom { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } = { A , B }
21	17, 20	syl6eqr 2491	. . . . 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 2508	. . . . . 6 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( x e. dom F <-> x e. { A , B } ) )
25		vex 2973	. . . . . . . 8 |- x e. _V
26	25	elpr 3893	. . . . . . 7 |- ( x e. { A , B } <-> ( x = A \/ x = B ) )
27	1, 18	fvpr1 5919	. . . . . . . . . . 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 2446	. . . . . . . . 9 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( F ` A ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` A ) )
30		fveq2 5689	. . . . . . . . . 10 |- ( x = A -> ( F ` x ) = ( F ` A ) )
31		fveq2 5689	. . . . . . . . . 10 |- ( x = A -> ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` A ) )
32	30, 31	eqeq12d 2455	. . . . . . . . 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 5920	. . . . . . . . . . 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 2446	. . . . . . . . 9 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( F ` B ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` B ) )
38		fveq2 5689	. . . . . . . . . 10 |- ( x = B -> ( F ` x ) = ( F ` B ) )
39		fveq2 5689	. . . . . . . . . 10 |- ( x = B -> ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` B ) )
40	38, 39	eqeq12d 2455	. . . . . . . . 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 2796	. . . 4 |- ( ( A =/= B /\ F Fn { A , B } ) -> A. x e. dom F ( F ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) )
46		fnfun 5506	. . . . 5 |- ( F Fn { A , B } -> Fun F )
47	1, 34, 18, 19	funpr 5467	. . . . 5 |- ( A =/= B -> Fun { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } )
48		eqfunfv 5800	. . . . 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 913	. . 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 5419	. . . . 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 5497	. . . . 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 828	. 2 |- ( A =/= B -> ( F Fn { A , B } <-> F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) )
58	16, 57	pm2.61ine 2685	1 |- ( F Fn { A , B } <-> F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } )

Syntax hints:   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1369   e. wcel 1756   =/= wne 2604   A.wral 2713   _Vcvv 2970   {csn 3875   {cpr 3877   <.cop 3881   dom cdm 4838   Fun wfun 5410   Fn wfn 5411   ` cfv 5416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424

This theorem is referenced by:  wrd2pr2op  12545