Theorem fnpr2g 6474

Description: A function whose domain has at most two elements can be represented as a set of at most two ordered pairs. (Contributed by Thierry Arnoux, 12-Jul-2020.)

Assertion

Ref	Expression
fnpr2g	|- ( ( A e. V /\ B e. W ) -> ( F Fn { A , B } <-> F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) )

Proof of Theorem fnpr2g

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		preq1 4268	. . . 4 |- ( a = A -> { a , b } = { A , b } )
2	1	fneq2d 5982	. . 3 |- ( a = A -> ( F Fn { a , b } <-> F Fn { A , b } ) )
3		id 22	. . . . . 6 |- ( a = A -> a = A )
4		fveq2 6191	. . . . . 6 |- ( a = A -> ( F ` a ) = ( F ` A ) )
5	3, 4	opeq12d 4410	. . . . 5 |- ( a = A -> <. a , ( F ` a ) >. = <. A , ( F ` A ) >. )
6	5	preq1d 4274	. . . 4 |- ( a = A -> { <. a , ( F ` a ) >. , <. b , ( F ` b ) >. } = { <. A , ( F ` A ) >. , <. b , ( F ` b ) >. } )
7	6	eqeq2d 2632	. . 3 |- ( a = A -> ( F = { <. a , ( F ` a ) >. , <. b , ( F ` b ) >. } <-> F = { <. A , ( F ` A ) >. , <. b , ( F ` b ) >. } ) )
8	2, 7	bibi12d 335	. 2 |- ( a = A -> ( ( F Fn { a , b } <-> F = { <. a , ( F ` a ) >. , <. b , ( F ` b ) >. } ) <-> ( F Fn { A , b } <-> F = { <. A , ( F ` A ) >. , <. b , ( F ` b ) >. } ) ) )
9		preq2 4269	. . . 4 |- ( b = B -> { A , b } = { A , B } )
10	9	fneq2d 5982	. . 3 |- ( b = B -> ( F Fn { A , b } <-> F Fn { A , B } ) )
11		id 22	. . . . . 6 |- ( b = B -> b = B )
12		fveq2 6191	. . . . . 6 |- ( b = B -> ( F ` b ) = ( F ` B ) )
13	11, 12	opeq12d 4410	. . . . 5 |- ( b = B -> <. b , ( F ` b ) >. = <. B , ( F ` B ) >. )
14	13	preq2d 4275	. . . 4 |- ( b = B -> { <. A , ( F ` A ) >. , <. b , ( F ` b ) >. } = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } )
15	14	eqeq2d 2632	. . 3 |- ( b = B -> ( F = { <. A , ( F ` A ) >. , <. b , ( F ` b ) >. } <-> F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) )
16	10, 15	bibi12d 335	. 2 |- ( b = B -> ( ( F Fn { A , b } <-> F = { <. A , ( F ` A ) >. , <. b , ( F ` b ) >. } ) <-> ( F Fn { A , B } <-> F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) ) )
17		vex 3203	. . 3 |- a e. _V
18		vex 3203	. . 3 |- b e. _V
19	17, 18	fnprb 6472	. 2 |- ( F Fn { a , b } <-> F = { <. a , ( F ` a ) >. , <. b , ( F ` b ) >. } )
20	8, 16, 19	vtocl2g 3270	1 |- ( ( A e. V /\ B e. W ) -> ( F Fn { A , B } <-> F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cpr 4179   <.cop 4183   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:  fpr2g  6475