Theorem fnprb 6134

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 6094	. . . . 5 |- ( F Fn { A } <-> F = { <. A , ( F ` A ) >. } )
3		dfsn2 4009	. . . . . 6 |- { A } = { A , A }
4	3	fneq2i 5685	. . . . 5 |- ( F Fn { A } <-> F Fn { A , A } )
5		dfsn2 4009	. . . . . 6 |- { <. A , ( F ` A ) >. } = { <. A , ( F ` A ) >. , <. A , ( F ` A ) >. }
6	5	eqeq2i 2440	. . . . 5 |- ( F = { <. A , ( F ` A ) >. } <-> F = { <. A , ( F ` A ) >. , <. A , ( F ` A ) >. } )
7	2, 4, 6	3bitr3i 278	. . . 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 4077	. . . 4 |- ( A = B -> { A , A } = { A , B } )
10	9	fneq2d 5681	. . 3 |- ( A = B -> ( F Fn { A , A } <-> F Fn { A , B } ) )
11		id 23	. . . . . 6 |- ( A = B -> A = B )
12		fveq2 5877	. . . . . 6 |- ( A = B -> ( F ` A ) = ( F ` B ) )
13	11, 12	opeq12d 4192	. . . . 5 |- ( A = B -> <. A , ( F ` A ) >. = <. B , ( F ` B ) >. )
14	13	preq2d 4083	. . . 4 |- ( A = B -> { <. A , ( F ` A ) >. , <. A , ( F ` A ) >. } = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } )
15	14	eqeq2d 2436	. . 3 |- ( A = B -> ( F = { <. A , ( F ` A ) >. , <. A , ( F ` A ) >. } <-> F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) )
16	8, 10, 15	3bitr3d 286	. 2 |- ( A = B -> ( F Fn { A , B } <-> F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) )
17		fndm 5689	. . . . . 6 |- ( F Fn { A , B } -> dom F = { A , B } )
18		fvex 5887	. . . . . . 7 |- ( F ` A ) e. _V
19		fvex 5887	. . . . . . 7 |- ( F ` B ) e. _V
20	18, 19	dmprop 5326	. . . . . 6 |- dom { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } = { A , B }
21	17, 20	syl6eqr 2481	. . . . 5 |- ( F Fn { A , B } -> dom F = dom { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } )
22	21	adantl 467	. . . 4 |- ( ( A =/= B /\ F Fn { A , B } ) -> dom F = dom { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } )
23	17	adantl 467	. . . . . . 7 |- ( ( A =/= B /\ F Fn { A , B } ) -> dom F = { A , B } )
24	23	eleq2d 2492	. . . . . 6 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( x e. dom F <-> x e. { A , B } ) )
25		vex 3084	. . . . . . . 8 |- x e. _V
26	25	elpr 4014	. . . . . . 7 |- ( x e. { A , B } <-> ( x = A \/ x = B ) )
27	1, 18	fvpr1 6118	. . . . . . . . . . 11 |- ( A =/= B -> ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` A ) = ( F ` A ) )
28	27	adantr 466	. . . . . . . . . 10 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` A ) = ( F ` A ) )
29	28	eqcomd 2430	. . . . . . . . 9 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( F ` A ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` A ) )
30		fveq2 5877	. . . . . . . . . 10 |- ( x = A -> ( F ` x ) = ( F ` A ) )
31		fveq2 5877	. . . . . . . . . 10 |- ( x = A -> ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` A ) )
32	30, 31	eqeq12d 2444	. . . . . . . . 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 225	. . . . . . . 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 6119	. . . . . . . . . . 11 |- ( A =/= B -> ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` B ) = ( F ` B ) )
36	35	adantr 466	. . . . . . . . . 10 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` B ) = ( F ` B ) )
37	36	eqcomd 2430	. . . . . . . . 9 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( F ` B ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` B ) )
38		fveq2 5877	. . . . . . . . . 10 |- ( x = B -> ( F ` x ) = ( F ` B ) )
39		fveq2 5877	. . . . . . . . . 10 |- ( x = B -> ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` B ) )
40	38, 39	eqeq12d 2444	. . . . . . . . 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 225	. . . . . . . 8 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( x = B -> ( F ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) ) )
42	33, 41	jaod 381	. . . . . . 7 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( ( x = A \/ x = B ) -> ( F ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) ) )
43	26, 42	syl5bi 220	. . . . . 6 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( x e. { A , B } -> ( F ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) ) )
44	24, 43	sylbid 218	. . . . 5 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( x e. dom F -> ( F ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) ) )
45	44	ralrimiv 2837	. . . 4 |- ( ( A =/= B /\ F Fn { A , B } ) -> A. x e. dom F ( F ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) )
46		fnfun 5687	. . . . 5 |- ( F Fn { A , B } -> Fun F )
47	1, 34, 18, 19	funpr 5648	. . . . 5 |- ( A =/= B -> Fun { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } )
48		eqfunfv 5992	. . . . 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 480	. . . 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 930	. . 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 5600	. . . . 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 668	. . . 4 |- ( A =/= B -> { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } Fn { A , B } )
54		fneq1 5678	. . . . 5 |- ( F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } -> ( F Fn { A , B } <-> { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } Fn { A , B } ) )
55	54	biimprd 226	. . . 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 471	. . 3 |- ( ( A =/= B /\ F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) -> F Fn { A , B } )
57	50, 56	impbida 840	. 2 |- ( A =/= B -> ( F Fn { A , B } <-> F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) )
58	16, 57	pm2.61ine 2737	1 |- ( F Fn { A , B } <-> F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } )

Syntax hints:   <-> wb 187   \/ wo 369   /\ wa 370   = wceq 1437   e. wcel 1868   =/= wne 2618   A.wral 2775   _Vcvv 3081   {csn 3996   {cpr 3998   <.cop 4002   dom cdm 4849   Fun wfun 5591   Fn wfn 5592   ` cfv 5597

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605

This theorem is referenced by:  fnpr2g  6135  wrd2pr2op  13010