Theorem 1stconst 5070

Description: The mapping of a restriction of the 1st function to a constant function.

Assertion

Ref	Expression
1stconst	|- (B e. C -> (1st |` (A X. {B})):(A X. {B})-1-1-onto->A)

Proof of Theorem 1stconst

Step	Hyp	Ref	Expression
1		dff1o3 4641	. 2 |- ((1st |` (A X. {B})):(A X. {B})-1-1-onto->A <-> ((1st |` (A X. {B})):(A X. {B})-onto->A /\ Fun `'(1st |` (A X. {B}))))
2		snnzg 3118	. . 3 |- (B e. C -> {B} =/= (/))
3		fo1stres 5036	. . 3 |- ({B} =/= (/) -> (1st |` (A X. {B})):(A X. {B})-onto->A)
4	2, 3	syl 12	. 2 |- (B e. C -> (1st |` (A X. {B})):(A X. {B})-onto->A)
5		moeq 2431	. . . . . 6 |- E*x x = <.y, B>.
6	5	moani 1820	. . . . 5 |- E*x(y e. A /\ x = <.y, B>.)
7		ax-17 1317	. . . . . 6 |- (B e. C -> A.x B e. C)
8		eleq1 1957	. . . . . . . . . . . . . . 15 |- ((1st` x) = y -> ((1st` x) e. A <-> y e. A))
9	8	biimpa 460	. . . . . . . . . . . . . 14 |- (((1st` x) = y /\ (1st` x) e. A) -> y e. A)
10	9	adantrr 431	. . . . . . . . . . . . 13 |- (((1st` x) = y /\ ((1st` x) e. A /\ (2nd` x) e. {B})) -> y e. A)
11	10	adantrl 430	. . . . . . . . . . . 12 |- (((1st` x) = y /\ (x e. (_V X. _V) /\ ((1st` x) e. A /\ (2nd` x) e. {B}))) -> y e. A)
12		eqopi 5043	. . . . . . . . . . . . . . 15 |- ((x e. (_V X. _V) /\ ((1st` x) = y /\ (2nd` x) = B)) -> x = <.y, B>.)
13	12	an1s 544	. . . . . . . . . . . . . 14 |- (((1st` x) = y /\ (x e. (_V X. _V) /\ (2nd` x) = B)) -> x = <.y, B>.)
14		elsni 3066	. . . . . . . . . . . . . 14 |- ((2nd` x) e. {B} -> (2nd` x) = B)
15	13, 14	sylanr2 512	. . . . . . . . . . . . 13 |- (((1st` x) = y /\ (x e. (_V X. _V) /\ (2nd` x) e. {B})) -> x = <.y, B>.)
16	15	adantrrl 438	. . . . . . . . . . . 12 |- (((1st` x) = y /\ (x e. (_V X. _V) /\ ((1st` x) e. A /\ (2nd` x) e. {B}))) -> x = <.y, B>.)
17	11, 16	jca 310	. . . . . . . . . . 11 |- (((1st` x) = y /\ (x e. (_V X. _V) /\ ((1st` x) e. A /\ (2nd` x) e. {B}))) -> (y e. A /\ x = <.y, B>.))
18		elxp7 5042	. . . . . . . . . . 11 |- (x e. (A X. {B}) <-> (x e. (_V X. _V) /\ ((1st` x) e. A /\ (2nd` x) e. {B})))
19	17, 18	sylan2b 501	. . . . . . . . . 10 |- (((1st` x) = y /\ x e. (A X. {B})) -> (y e. A /\ x = <.y, B>.))
20	19	adantl 424	. . . . . . . . 9 |- ((B e. C /\ ((1st` x) = y /\ x e. (A X. {B}))) -> (y e. A /\ x = <.y, B>.))
21		fveq2 4681	. . . . . . . . . . . 12 |- (x = <.y, B>. -> (1st` x) = (1st` <.y, B>.))
22		visset 2295	. . . . . . . . . . . . 13 |- y e. _V
23	22	op1st 5026	. . . . . . . . . . . 12 |- (1st` <.y, B>.) = y
24	21, 23	syl6eq 1944	. . . . . . . . . . 11 |- (x = <.y, B>. -> (1st` x) = y)
25	24	ad2antll 443	. . . . . . . . . 10 |- ((B e. C /\ (y e. A /\ x = <.y, B>.)) -> (1st` x) = y)
26		simprr 451	. . . . . . . . . . 11 |- ((B e. C /\ (y e. A /\ x = <.y, B>.)) -> x = <.y, B>.)
27		simprl 450	. . . . . . . . . . . 12 |- ((B e. C /\ (y e. A /\ x = <.y, B>.)) -> y e. A)
28		snidg 3067	. . . . . . . . . . . . 13 |- (B e. C -> B e. {B})
29	28	adantr 425	. . . . . . . . . . . 12 |- ((B e. C /\ (y e. A /\ x = <.y, B>.)) -> B e. {B})
30		opelxpi 4040	. . . . . . . . . . . 12 |- ((y e. A /\ B e. {B}) -> <.y, B>. e. (A X. {B}))
31	27, 29, 30	syl11anc 524	. . . . . . . . . . 11 |- ((B e. C /\ (y e. A /\ x = <.y, B>.)) -> <.y, B>. e. (A X. {B}))
32	26, 31	eqeltrd 1971	. . . . . . . . . 10 |- ((B e. C /\ (y e. A /\ x = <.y, B>.)) -> x e. (A X. {B}))
33	25, 32	jca 310	. . . . . . . . 9 |- ((B e. C /\ (y e. A /\ x = <.y, B>.)) -> ((1st` x) = y /\ x e. (A X. {B})))
34	20, 33	impbida 577	. . . . . . . 8 |- (B e. C -> (((1st` x) = y /\ x e. (A X. {B})) <-> (y e. A /\ x = <.y, B>.)))
35		fo1st 5032	. . . . . . . . . . 11 |- 1st:_V-onto->_V
36		fofn 4619	. . . . . . . . . . 11 |- (1st:_V-onto->_V -> 1st Fn _V)
37	35, 36	ax-mp 7	. . . . . . . . . 10 |- 1st Fn _V
38		visset 2295	. . . . . . . . . 10 |- x e. _V
39	22	fnbrfvb 4712	. . . . . . . . . 10 |- ((1st Fn _V /\ x e. _V) -> ((1st` x) = y <-> x1sty))
40	37, 38, 39	mp2an 761	. . . . . . . . 9 |- ((1st` x) = y <-> x1sty)
41	40	anbi1i 539	. . . . . . . 8 |- (((1st` x) = y /\ x e. (A X. {B})) <-> (x1sty /\ x e. (A X. {B})))
42	34, 41	syl5bbr 593	. . . . . . 7 |- (B e. C -> ((x1sty /\ x e. (A X. {B})) <-> (y e. A /\ x = <.y, B>.)))
43	22	brres 4223	. . . . . . 7 |- (x(1st |` (A X. {B}))y <-> (x1sty /\ x e. (A X. {B})))
44	42, 43	syl5bb 591	. . . . . 6 |- (B e. C -> (x(1st |` (A X. {B}))y <-> (y e. A /\ x = <.y, B>.)))
45	7, 44	mobid 1800	. . . . 5 |- (B e. C -> (E*x x(1st |` (A X. {B}))y <-> E*x(y e. A /\ x = <.y, B>.)))
46	6, 45	mpbiri 211	. . . 4 |- (B e. C -> E*x x(1st |` (A X. {B}))y)
47	46	19.21aiv 1664	. . 3 |- (B e. C -> A.yE*x x(1st |` (A X. {B}))y)
48		funcnv2 4474	. . 3 |- (Fun `'(1st |` (A X. {B})) <-> A.yE*x x(1st |` (A X. {B}))y)
49	47, 48	sylibr 217	. 2 |- (B e. C -> Fun `'(1st |` (A X. {B})))
50	1, 4, 49	sylanbrc 527	1 |- (B e. C -> (1st |` (A X. {B})):(A X. {B})-1-1-onto->A)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E*wmo 1772   =/= wne 2017  _Vcvv 2292  (/)c0 2875  {csn 3044  <.cop 3046   class class class wbr 3338   X. cxp 3984  `'ccnv 3985   |` cres 3988  Fun wfun 3992   Fn wfn 3993  -onto->wfo 3996  -1-1-onto->wf1o 3997  ` cfv 3998  1stc1st 5018  2ndc2nd 5019

This theorem is referenced by:  curry2 5078

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-1st 5020  df-2nd 5021

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