Theorem fsets 15891

Description: The structure replacement function is a function. (Contributed by SO, 12-Jul-2018.)

Assertion

Ref	Expression
fsets	|- ( ( ( F e. V /\ F : A --> B ) /\ X e. A /\ Y e. B ) -> ( F sSet <. X , Y >. ) : A --> B )

Proof of Theorem fsets

Step	Hyp	Ref	Expression
1		difss 3737	. . . . . 6 |- ( A \ { X } ) C_ A
2		fssres 6070	. . . . . 6 |- ( ( F : A --> B /\ ( A \ { X } ) C_ A ) -> ( F |` ( A \ { X } ) ) : ( A \ { X } ) --> B )
3	1, 2	mpan2 707	. . . . 5 |- ( F : A --> B -> ( F |` ( A \ { X } ) ) : ( A \ { X } ) --> B )
4		resres 5409	. . . . . . . 8 |- ( ( F |` A ) |` ( _V \ { X } ) ) = ( F |` ( A i^i ( _V \ { X } ) ) )
5		invdif 3868	. . . . . . . . 9 |- ( A i^i ( _V \ { X } ) ) = ( A \ { X } )
6	5	reseq2i 5393	. . . . . . . 8 |- ( F |` ( A i^i ( _V \ { X } ) ) ) = ( F |` ( A \ { X } ) )
7	4, 6	eqtri 2644	. . . . . . 7 |- ( ( F |` A ) |` ( _V \ { X } ) ) = ( F |` ( A \ { X } ) )
8		ffn 6045	. . . . . . . . 9 |- ( F : A --> B -> F Fn A )
9		fnresdm 6000	. . . . . . . . 9 |- ( F Fn A -> ( F |` A ) = F )
10	8, 9	syl 17	. . . . . . . 8 |- ( F : A --> B -> ( F |` A ) = F )
11	10	reseq1d 5395	. . . . . . 7 |- ( F : A --> B -> ( ( F |` A ) |` ( _V \ { X } ) ) = ( F |` ( _V \ { X } ) ) )
12	7, 11	syl5reqr 2671	. . . . . 6 |- ( F : A --> B -> ( F |` ( _V \ { X } ) ) = ( F |` ( A \ { X } ) ) )
13	12	feq1d 6030	. . . . 5 |- ( F : A --> B -> ( ( F |` ( _V \ { X } ) ) : ( A \ { X } ) --> B <-> ( F |` ( A \ { X } ) ) : ( A \ { X } ) --> B ) )
14	3, 13	mpbird 247	. . . 4 |- ( F : A --> B -> ( F |` ( _V \ { X } ) ) : ( A \ { X } ) --> B )
15	14	adantl 482	. . 3 |- ( ( F e. V /\ F : A --> B ) -> ( F |` ( _V \ { X } ) ) : ( A \ { X } ) --> B )
16		fsnunf2 6452	. . 3 |- ( ( ( F |` ( _V \ { X } ) ) : ( A \ { X } ) --> B /\ X e. A /\ Y e. B ) -> ( ( F |` ( _V \ { X } ) ) u. { <. X , Y >. } ) : A --> B )
17	15, 16	syl3an1 1359	. 2 |- ( ( ( F e. V /\ F : A --> B ) /\ X e. A /\ Y e. B ) -> ( ( F |` ( _V \ { X } ) ) u. { <. X , Y >. } ) : A --> B )
18		simp1l 1085	. . 3 |- ( ( ( F e. V /\ F : A --> B ) /\ X e. A /\ Y e. B ) -> F e. V )
19		simp3 1063	. . 3 |- ( ( ( F e. V /\ F : A --> B ) /\ X e. A /\ Y e. B ) -> Y e. B )
20		setsval 15888	. . . 4 |- ( ( F e. V /\ Y e. B ) -> ( F sSet <. X , Y >. ) = ( ( F |` ( _V \ { X } ) ) u. { <. X , Y >. } ) )
21	20	feq1d 6030	. . 3 |- ( ( F e. V /\ Y e. B ) -> ( ( F sSet <. X , Y >. ) : A --> B <-> ( ( F |` ( _V \ { X } ) ) u. { <. X , Y >. } ) : A --> B ) )
22	18, 19, 21	syl2anc 693	. 2 |- ( ( ( F e. V /\ F : A --> B ) /\ X e. A /\ Y e. B ) -> ( ( F sSet <. X , Y >. ) : A --> B <-> ( ( F |` ( _V \ { X } ) ) u. { <. X , Y >. } ) : A --> B ) )
23	17, 22	mpbird 247	1 |- ( ( ( F e. V /\ F : A --> B ) /\ X e. A /\ Y e. B ) -> ( F sSet <. X , Y >. ) : A --> B )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   {csn 4177   <.cop 4183   |` cres 5116   Fn wfn 5883   -->wf 5884  (class class class)co 6650   sSet csts 15855

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-pr 4906  ax-un 6949

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-rab 2921  df-v 3202  df-sbc 3436  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-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-sets 15864

This theorem is referenced by:  mdetunilem9  20426