Theorem xpsc0 16220

Description: The pair function maps 0 to A. (Contributed by Mario Carneiro, 14-Aug-2015.)

Assertion

Ref	Expression
xpsc0	|- ( A e. V -> ( `' ( { A } +c { B } ) ` (/) ) = A )

Proof of Theorem xpsc0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpsc 16217	. . . 4 |- `' ( { A } +c { B } ) = ( ( { (/) } X. { A } ) u. ( { 1o } X. { B } ) )
2	1	fveq1i 6192	. . 3 |- ( `' ( { A } +c { B } ) ` (/) ) = ( ( ( { (/) } X. { A } ) u. ( { 1o } X. { B } ) ) ` (/) )
3		fnconstg 6093	. . . 4 |- ( A e. V -> ( { (/) } X. { A } ) Fn { (/) } )
4		vex 3203	. . . . . . . . . . . . 13 |- x e. _V
5		fvi 6255	. . . . . . . . . . . . 13 |- ( x e. _V -> ( _I ` x ) = x )
6	4, 5	ax-mp 5	. . . . . . . . . . . 12 |- ( _I ` x ) = x
7		elsni 4194	. . . . . . . . . . . . 13 |- ( x e. { B } -> x = B )
8	7	fveq2d 6195	. . . . . . . . . . . 12 |- ( x e. { B } -> ( _I ` x ) = ( _I ` B ) )
9	6, 8	syl5eqr 2670	. . . . . . . . . . 11 |- ( x e. { B } -> x = ( _I ` B ) )
10		velsn 4193	. . . . . . . . . . 11 |- ( x e. { ( _I ` B ) } <-> x = ( _I ` B ) )
11	9, 10	sylibr 224	. . . . . . . . . 10 |- ( x e. { B } -> x e. { ( _I ` B ) } )
12	11	ssriv 3607	. . . . . . . . 9 |- { B } C_ { ( _I ` B ) }
13		xpss2 5229	. . . . . . . . 9 |- ( { B } C_ { ( _I ` B ) } -> ( { 1o } X. { B } ) C_ ( { 1o } X. { ( _I ` B ) } ) )
14	12, 13	ax-mp 5	. . . . . . . 8 |- ( { 1o } X. { B } ) C_ ( { 1o } X. { ( _I ` B ) } )
15		1on 7567	. . . . . . . . . 10 |- 1o e. On
16	15	elexi 3213	. . . . . . . . 9 |- 1o e. _V
17		fvex 6201	. . . . . . . . 9 |- ( _I ` B ) e. _V
18	16, 17	xpsn 6407	. . . . . . . 8 |- ( { 1o } X. { ( _I ` B ) } ) = { <. 1o , ( _I ` B ) >. }
19	14, 18	sseqtri 3637	. . . . . . 7 |- ( { 1o } X. { B } ) C_ { <. 1o , ( _I ` B ) >. }
20	16, 17	funsn 5939	. . . . . . 7 |- Fun { <. 1o , ( _I ` B ) >. }
21		funss 5907	. . . . . . 7 |- ( ( { 1o } X. { B } ) C_ { <. 1o , ( _I ` B ) >. } -> ( Fun { <. 1o , ( _I ` B ) >. } -> Fun ( { 1o } X. { B } ) ) )
22	19, 20, 21	mp2 9	. . . . . 6 |- Fun ( { 1o } X. { B } )
23		funfn 5918	. . . . . 6 |- ( Fun ( { 1o } X. { B } ) <-> ( { 1o } X. { B } ) Fn dom ( { 1o } X. { B } ) )
24	22, 23	mpbi 220	. . . . 5 |- ( { 1o } X. { B } ) Fn dom ( { 1o } X. { B } )
25	24	a1i 11	. . . 4 |- ( A e. V -> ( { 1o } X. { B } ) Fn dom ( { 1o } X. { B } ) )
26		dmxpss 5565	. . . . . . 7 |- dom ( { 1o } X. { B } ) C_ { 1o }
27		sslin 3839	. . . . . . 7 |- ( dom ( { 1o } X. { B } ) C_ { 1o } -> ( { (/) } i^i dom ( { 1o } X. { B } ) ) C_ ( { (/) } i^i { 1o } ) )
28	26, 27	ax-mp 5	. . . . . 6 |- ( { (/) } i^i dom ( { 1o } X. { B } ) ) C_ ( { (/) } i^i { 1o } )
29		1n0 7575	. . . . . . . 8 |- 1o =/= (/)
30	29	necomi 2848	. . . . . . 7 |- (/) =/= 1o
31		disjsn2 4247	. . . . . . 7 |- ( (/) =/= 1o -> ( { (/) } i^i { 1o } ) = (/) )
32	30, 31	ax-mp 5	. . . . . 6 |- ( { (/) } i^i { 1o } ) = (/)
33		sseq0 3975	. . . . . 6 |- ( ( ( { (/) } i^i dom ( { 1o } X. { B } ) ) C_ ( { (/) } i^i { 1o } ) /\ ( { (/) } i^i { 1o } ) = (/) ) -> ( { (/) } i^i dom ( { 1o } X. { B } ) ) = (/) )
34	28, 32, 33	mp2an 708	. . . . 5 |- ( { (/) } i^i dom ( { 1o } X. { B } ) ) = (/)
35	34	a1i 11	. . . 4 |- ( A e. V -> ( { (/) } i^i dom ( { 1o } X. { B } ) ) = (/) )
36		0ex 4790	. . . . . 6 |- (/) e. _V
37	36	snid 4208	. . . . 5 |- (/) e. { (/) }
38	37	a1i 11	. . . 4 |- ( A e. V -> (/) e. { (/) } )
39		fvun1 6269	. . . 4 |- ( ( ( { (/) } X. { A } ) Fn { (/) } /\ ( { 1o } X. { B } ) Fn dom ( { 1o } X. { B } ) /\ ( ( { (/) } i^i dom ( { 1o } X. { B } ) ) = (/) /\ (/) e. { (/) } ) ) -> ( ( ( { (/) } X. { A } ) u. ( { 1o } X. { B } ) ) ` (/) ) = ( ( { (/) } X. { A } ) ` (/) ) )
40	3, 25, 35, 38, 39	syl112anc 1330	. . 3 |- ( A e. V -> ( ( ( { (/) } X. { A } ) u. ( { 1o } X. { B } ) ) ` (/) ) = ( ( { (/) } X. { A } ) ` (/) ) )
41	2, 40	syl5eq 2668	. 2 |- ( A e. V -> ( `' ( { A } +c { B } ) ` (/) ) = ( ( { (/) } X. { A } ) ` (/) ) )
42		xpsng 6406	. . . . 5 |- ( ( (/) e. _V /\ A e. V ) -> ( { (/) } X. { A } ) = { <. (/) , A >. } )
43	42	fveq1d 6193	. . . 4 |- ( ( (/) e. _V /\ A e. V ) -> ( ( { (/) } X. { A } ) ` (/) ) = ( { <. (/) , A >. } ` (/) ) )
44		fvsng 6447	. . . 4 |- ( ( (/) e. _V /\ A e. V ) -> ( { <. (/) , A >. } ` (/) ) = A )
45	43, 44	eqtrd 2656	. . 3 |- ( ( (/) e. _V /\ A e. V ) -> ( ( { (/) } X. { A } ) ` (/) ) = A )
46	36, 45	mpan 706	. 2 |- ( A e. V -> ( ( { (/) } X. { A } ) ` (/) ) = A )
47	41, 46	eqtrd 2656	1 |- ( A e. V -> ( `' ( { A } +c { B } ) ` (/) ) = A )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183   _I cid 5023   X. cxp 5112   `'ccnv 5113   dom cdm 5114   Oncon0 5723   Fun wfun 5882   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   1oc1o 7553   +c ccda 8989

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  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-ord 5726  df-on 5727  df-suc 5729  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-1o 7560  df-cda 8990

This theorem is referenced by:  xpscfv  16222  xpsfeq  16224  xpsfrnel2  16225  xpsff1o  16228  xpsle  16241  dmdprdpr  18448  dprdpr  18449  xpstopnlem1  21612  xpstopnlem2  21614  xpsxmetlem  22184  xpsdsval  22186  xpsmet  22187