Theorem 2ndconst 5071

Description: The mapping of a restriction of the 2nd function to a converse constant function.

Assertion

Ref	Expression
2ndconst	|- (A e. C -> (2nd |` ({A} X. B)):({A} X. B)-1-1-onto->B)

Proof of Theorem 2ndconst

Step	Hyp	Ref	Expression
1		dff1o3 4641	. 2 |- ((2nd |` ({A} X. B)):({A} X. B)-1-1-onto->B <-> ((2nd |` ({A} X. B)):({A} X. B)-onto->B /\ Fun `'(2nd |` ({A} X. B))))
2		snnzg 3118	. . 3 |- (A e. C -> {A} =/= (/))
3		fo2ndres 5037	. . 3 |- ({A} =/= (/) -> (2nd |` ({A} X. B)):({A} X. B)-onto->B)
4	2, 3	syl 12	. 2 |- (A e. C -> (2nd |` ({A} X. B)):({A} X. B)-onto->B)
5		moeq 2431	. . . . . 6 |- E*x x = <.A, y>.
6	5	moani 1820	. . . . 5 |- E*x(y e. B /\ x = <.A, y>.)
7		ax-17 1317	. . . . . 6 |- (A e. C -> A.x A e. C)
8		eleq1 1957	. . . . . . . . . . . . . . 15 |- ((2nd` x) = y -> ((2nd` x) e. B <-> y e. B))
9	8	biimpa 460	. . . . . . . . . . . . . 14 |- (((2nd` x) = y /\ (2nd` x) e. B) -> y e. B)
10	9	adantrl 430	. . . . . . . . . . . . 13 |- (((2nd` x) = y /\ ((1st` x) e. {A} /\ (2nd` x) e. B)) -> y e. B)
11	10	adantrl 430	. . . . . . . . . . . 12 |- (((2nd` x) = y /\ (x e. (_V X. _V) /\ ((1st` x) e. {A} /\ (2nd` x) e. B))) -> y e. B)
12		eqopi 5043	. . . . . . . . . . . . . . . 16 |- ((x e. (_V X. _V) /\ ((1st` x) = A /\ (2nd` x) = y)) -> x = <.A, y>.)
13	12	ancom2s 545	. . . . . . . . . . . . . . 15 |- ((x e. (_V X. _V) /\ ((2nd` x) = y /\ (1st` x) = A)) -> x = <.A, y>.)
14	13	an1s 544	. . . . . . . . . . . . . 14 |- (((2nd` x) = y /\ (x e. (_V X. _V) /\ (1st` x) = A)) -> x = <.A, y>.)
15		elsni 3066	. . . . . . . . . . . . . 14 |- ((1st` x) e. {A} -> (1st` x) = A)
16	14, 15	sylanr2 512	. . . . . . . . . . . . 13 |- (((2nd` x) = y /\ (x e. (_V X. _V) /\ (1st` x) e. {A})) -> x = <.A, y>.)
17	16	adantrrr 439	. . . . . . . . . . . 12 |- (((2nd` x) = y /\ (x e. (_V X. _V) /\ ((1st` x) e. {A} /\ (2nd` x) e. B))) -> x = <.A, y>.)
18	11, 17	jca 310	. . . . . . . . . . 11 |- (((2nd` x) = y /\ (x e. (_V X. _V) /\ ((1st` x) e. {A} /\ (2nd` x) e. B))) -> (y e. B /\ x = <.A, y>.))
19		elxp7 5042	. . . . . . . . . . 11 |- (x e. ({A} X. B) <-> (x e. (_V X. _V) /\ ((1st` x) e. {A} /\ (2nd` x) e. B)))
20	18, 19	sylan2b 501	. . . . . . . . . 10 |- (((2nd` x) = y /\ x e. ({A} X. B)) -> (y e. B /\ x = <.A, y>.))
21	20	adantl 424	. . . . . . . . 9 |- ((A e. C /\ ((2nd` x) = y /\ x e. ({A} X. B))) -> (y e. B /\ x = <.A, y>.))
22		fveq2 4681	. . . . . . . . . . . 12 |- (x = <.A, y>. -> (2nd` x) = (2nd` <.A, y>.))
23		visset 2295	. . . . . . . . . . . . 13 |- y e. _V
24		op2ndg 5029	. . . . . . . . . . . . 13 |- ((A e. C /\ y e. _V) -> (2nd` <.A, y>.) = y)
25	23, 24	mpan2 760	. . . . . . . . . . . 12 |- (A e. C -> (2nd` <.A, y>.) = y)
26	22, 25	sylan9eqr 1951	. . . . . . . . . . 11 |- ((A e. C /\ x = <.A, y>.) -> (2nd` x) = y)
27	26	adantrl 430	. . . . . . . . . 10 |- ((A e. C /\ (y e. B /\ x = <.A, y>.)) -> (2nd` x) = y)
28		simprr 451	. . . . . . . . . . 11 |- ((A e. C /\ (y e. B /\ x = <.A, y>.)) -> x = <.A, y>.)
29		snidg 3067	. . . . . . . . . . . . 13 |- (A e. C -> A e. {A})
30	29	adantr 425	. . . . . . . . . . . 12 |- ((A e. C /\ (y e. B /\ x = <.A, y>.)) -> A e. {A})
31		simprl 450	. . . . . . . . . . . 12 |- ((A e. C /\ (y e. B /\ x = <.A, y>.)) -> y e. B)
32		opelxpi 4040	. . . . . . . . . . . 12 |- ((A e. {A} /\ y e. B) -> <.A, y>. e. ({A} X. B))
33	30, 31, 32	syl11anc 524	. . . . . . . . . . 11 |- ((A e. C /\ (y e. B /\ x = <.A, y>.)) -> <.A, y>. e. ({A} X. B))
34	28, 33	eqeltrd 1971	. . . . . . . . . 10 |- ((A e. C /\ (y e. B /\ x = <.A, y>.)) -> x e. ({A} X. B))
35	27, 34	jca 310	. . . . . . . . 9 |- ((A e. C /\ (y e. B /\ x = <.A, y>.)) -> ((2nd` x) = y /\ x e. ({A} X. B)))
36	21, 35	impbida 577	. . . . . . . 8 |- (A e. C -> (((2nd` x) = y /\ x e. ({A} X. B)) <-> (y e. B /\ x = <.A, y>.)))
37		fo2nd 5033	. . . . . . . . . . 11 |- 2nd:_V-onto->_V
38		fofn 4619	. . . . . . . . . . 11 |- (2nd:_V-onto->_V -> 2nd Fn _V)
39	37, 38	ax-mp 7	. . . . . . . . . 10 |- 2nd Fn _V
40		visset 2295	. . . . . . . . . 10 |- x e. _V
41	23	fnbrfvb 4712	. . . . . . . . . 10 |- ((2nd Fn _V /\ x e. _V) -> ((2nd` x) = y <-> x2ndy))
42	39, 40, 41	mp2an 761	. . . . . . . . 9 |- ((2nd` x) = y <-> x2ndy)
43	42	anbi1i 539	. . . . . . . 8 |- (((2nd` x) = y /\ x e. ({A} X. B)) <-> (x2ndy /\ x e. ({A} X. B)))
44	36, 43	syl5bbr 593	. . . . . . 7 |- (A e. C -> ((x2ndy /\ x e. ({A} X. B)) <-> (y e. B /\ x = <.A, y>.)))
45	23	brres 4223	. . . . . . 7 |- (x(2nd |` ({A} X. B))y <-> (x2ndy /\ x e. ({A} X. B)))
46	44, 45	syl5bb 591	. . . . . 6 |- (A e. C -> (x(2nd |` ({A} X. B))y <-> (y e. B /\ x = <.A, y>.)))
47	7, 46	mobid 1800	. . . . 5 |- (A e. C -> (E*x x(2nd |` ({A} X. B))y <-> E*x(y e. B /\ x = <.A, y>.)))
48	6, 47	mpbiri 211	. . . 4 |- (A e. C -> E*x x(2nd |` ({A} X. B))y)
49	48	19.21aiv 1664	. . 3 |- (A e. C -> A.yE*x x(2nd |` ({A} X. B))y)
50		funcnv2 4474	. . 3 |- (Fun `'(2nd |` ({A} X. B)) <-> A.yE*x x(2nd |` ({A} X. B))y)
51	49, 50	sylibr 217	. 2 |- (A e. C -> Fun `'(2nd |` ({A} X. B)))
52	1, 4, 51	sylanbrc 527	1 |- (A e. C -> (2nd |` ({A} X. B)):({A} X. B)-1-1-onto->B)

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:  curry1 5075  fixp 10180  domrancur1b 14548  domrancur1c 14550

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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