Theorem fnprb 6472

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.a	|- A e. _V
fnprb.b	|- 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.a	. . . . . 6 |- A e. _V
2	1	fnsnb 6432	. . . . 5 |- ( F Fn { A } <-> F = { <. A , ( F ` A ) >. } )
3		dfsn2 4190	. . . . . 6 |- { A } = { A , A }
4	3	fneq2i 5986	. . . . 5 |- ( F Fn { A } <-> F Fn { A , A } )
5		dfsn2 4190	. . . . . 6 |- { <. A , ( F ` A ) >. } = { <. A , ( F ` A ) >. , <. A , ( F ` A ) >. }
6	5	eqeq2i 2634	. . . . 5 |- ( F = { <. A , ( F ` A ) >. } <-> F = { <. A , ( F ` A ) >. , <. A , ( F ` A ) >. } )
7	2, 4, 6	3bitr3i 290	. . . 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 4269	. . . 4 |- ( A = B -> { A , A } = { A , B } )
10	9	fneq2d 5982	. . 3 |- ( A = B -> ( F Fn { A , A } <-> F Fn { A , B } ) )
11		id 22	. . . . . 6 |- ( A = B -> A = B )
12		fveq2 6191	. . . . . 6 |- ( A = B -> ( F ` A ) = ( F ` B ) )
13	11, 12	opeq12d 4410	. . . . 5 |- ( A = B -> <. A , ( F ` A ) >. = <. B , ( F ` B ) >. )
14	13	preq2d 4275	. . . 4 |- ( A = B -> { <. A , ( F ` A ) >. , <. A , ( F ` A ) >. } = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } )
15	14	eqeq2d 2632	. . 3 |- ( A = B -> ( F = { <. A , ( F ` A ) >. , <. A , ( F ` A ) >. } <-> F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) )
16	8, 10, 15	3bitr3d 298	. 2 |- ( A = B -> ( F Fn { A , B } <-> F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) )
17		fndm 5990	. . . . . 6 |- ( F Fn { A , B } -> dom F = { A , B } )
18		fvex 6201	. . . . . . 7 |- ( F ` A ) e. _V
19		fvex 6201	. . . . . . 7 |- ( F ` B ) e. _V
20	18, 19	dmprop 5610	. . . . . 6 |- dom { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } = { A , B }
21	17, 20	syl6eqr 2674	. . . . 5 |- ( F Fn { A , B } -> dom F = dom { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } )
22	21	adantl 482	. . . 4 |- ( ( A =/= B /\ F Fn { A , B } ) -> dom F = dom { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } )
23	17	adantl 482	. . . . . . 7 |- ( ( A =/= B /\ F Fn { A , B } ) -> dom F = { A , B } )
24	23	eleq2d 2687	. . . . . 6 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( x e. dom F <-> x e. { A , B } ) )
25		vex 3203	. . . . . . . 8 |- x e. _V
26	25	elpr 4198	. . . . . . 7 |- ( x e. { A , B } <-> ( x = A \/ x = B ) )
27	1, 18	fvpr1 6456	. . . . . . . . . . 11 |- ( A =/= B -> ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` A ) = ( F ` A ) )
28	27	adantr 481	. . . . . . . . . 10 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` A ) = ( F ` A ) )
29	28	eqcomd 2628	. . . . . . . . 9 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( F ` A ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` A ) )
30		fveq2 6191	. . . . . . . . . 10 |- ( x = A -> ( F ` x ) = ( F ` A ) )
31		fveq2 6191	. . . . . . . . . 10 |- ( x = A -> ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` A ) )
32	30, 31	eqeq12d 2637	. . . . . . . . 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 237	. . . . . . . 8 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( x = A -> ( F ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) ) )
34		fnprb.b	. . . . . . . . . . . 12 |- B e. _V
35	34, 19	fvpr2 6457	. . . . . . . . . . 11 |- ( A =/= B -> ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` B ) = ( F ` B ) )
36	35	adantr 481	. . . . . . . . . 10 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` B ) = ( F ` B ) )
37	36	eqcomd 2628	. . . . . . . . 9 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( F ` B ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` B ) )
38		fveq2 6191	. . . . . . . . . 10 |- ( x = B -> ( F ` x ) = ( F ` B ) )
39		fveq2 6191	. . . . . . . . . 10 |- ( x = B -> ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` B ) )
40	38, 39	eqeq12d 2637	. . . . . . . . 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 237	. . . . . . . 8 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( x = B -> ( F ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) ) )
42	33, 41	jaod 395	. . . . . . 7 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( ( x = A \/ x = B ) -> ( F ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) ) )
43	26, 42	syl5bi 232	. . . . . 6 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( x e. { A , B } -> ( F ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) ) )
44	24, 43	sylbid 230	. . . . 5 |- ( ( A =/= B /\ F Fn { A , B } ) -> ( x e. dom F -> ( F ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) ) )
45	44	ralrimiv 2965	. . . 4 |- ( ( A =/= B /\ F Fn { A , B } ) -> A. x e. dom F ( F ` x ) = ( { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ` x ) )
46		fnfun 5988	. . . . 5 |- ( F Fn { A , B } -> Fun F )
47	1, 34, 18, 19	funpr 5944	. . . . 5 |- ( A =/= B -> Fun { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } )
48		eqfunfv 6316	. . . . 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 495	. . . 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 957	. . 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 5891	. . . . 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 698	. . . 4 |- ( A =/= B -> { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } Fn { A , B } )
54		fneq1 5979	. . . . 5 |- ( F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } -> ( F Fn { A , B } <-> { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } Fn { A , B } ) )
55	54	biimprd 238	. . . 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 486	. . 3 |- ( ( A =/= B /\ F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) -> F Fn { A , B } )
57	50, 56	impbida 877	. 2 |- ( A =/= B -> ( F Fn { A , B } <-> F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) )
58	16, 57	pm2.61ine 2877	1 |- ( F Fn { A , B } <-> F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } )

Syntax hints:   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   {csn 4177   {cpr 4179   <.cop 4183   dom cdm 5114   Fun wfun 5882   Fn wfn 5883   ` 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-reu 2919  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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by:  fntpb  6473  fnpr2g  6474  wrd2pr2op  13687