Theorem enprmaplem1 6076

Description: Lemma for enprmap 6082. Set up stratification. (Contributed by SF, 3-Mar-2015.)

Hypothesis

Ref	Expression
enprmaplem1.1	|- W = (r e. (A ^m B) |-> (`'r"{x}))

Assertion

Ref	Expression
enprmaplem1	|- W e. _V

Distinct variable groups:   A,r   B,r   x,r

Allowed substitution hints:   A(x)   B(x)   W(x,r)

Proof of Theorem enprmaplem1

Dummy variables y t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		enprmaplem1.1	. . 3 |- W = (r e. (A ^m B) |-> (`'r"{x}))
2		elima1c 4947	. . . . . . 7 |- (<.{y}, r>. e. (( SI (1st |` (`'2nd"{x})) (x) S )"1c) <-> E.t<.{t}, <.{y}, r>.>. e. ( SI (1st |` (`'2nd"{x})) (x) S ))
3		oteltxp 5782	. . . . . . . . 9 |- (<.{t}, <.{y}, r>.>. e. ( SI (1st |` (`'2nd"{x})) (x) S ) <-> (<.{t}, {y}>. e. SI (1st |` (`'2nd"{x})) /\ <.{t}, r>. e. S ))
4		vex 2862	. . . . . . . . . . . 12 |- t e. _V
5		vex 2862	. . . . . . . . . . . 12 |- y e. _V
6	4, 5	opsnelsi 5774	. . . . . . . . . . 11 |- (<.{t}, {y}>. e. SI (1st |` (`'2nd"{x})) <-> <.t, y>. e. (1st |` (`'2nd"{x})))
7		df-br 4640	. . . . . . . . . . . 12 |- (t(1st |` (`'2nd"{x}))y <-> <.t, y>. e. (1st |` (`'2nd"{x})))
8		brres 4949	. . . . . . . . . . . . 13 |- (t(1st |` (`'2nd"{x}))y <-> (t1sty /\ t e. (`'2nd"{x})))
9		eliniseg 5020	. . . . . . . . . . . . . 14 |- (t e. (`'2nd"{x}) <-> t2ndx)
10	9	anbi2i 675	. . . . . . . . . . . . 13 |- ((t1sty /\ t e. (`'2nd"{x})) <-> (t1sty /\ t2ndx))
11	8, 10	bitri 240	. . . . . . . . . . . 12 |- (t(1st |` (`'2nd"{x}))y <-> (t1sty /\ t2ndx))
12	7, 11	bitr3i 242	. . . . . . . . . . 11 |- (<.t, y>. e. (1st |` (`'2nd"{x})) <-> (t1sty /\ t2ndx))
13		vex 2862	. . . . . . . . . . . 12 |- x e. _V
14	5, 13	op1st2nd 5790	. . . . . . . . . . 11 |- ((t1sty /\ t2ndx) <-> t = <.y, x>.)
15	6, 12, 14	3bitri 262	. . . . . . . . . 10 |- (<.{t}, {y}>. e. SI (1st |` (`'2nd"{x})) <-> t = <.y, x>.)
16		vex 2862	. . . . . . . . . . 11 |- r e. _V
17	4, 16	opelssetsn 4760	. . . . . . . . . 10 |- (<.{t}, r>. e. S <-> t e. r)
18	15, 17	anbi12i 678	. . . . . . . . 9 |- ((<.{t}, {y}>. e. SI (1st |` (`'2nd"{x})) /\ <.{t}, r>. e. S ) <-> (t = <.y, x>. /\ t e. r))
19	3, 18	bitri 240	. . . . . . . 8 |- (<.{t}, <.{y}, r>.>. e. ( SI (1st |` (`'2nd"{x})) (x) S ) <-> (t = <.y, x>. /\ t e. r))
20	19	exbii 1582	. . . . . . 7 |- (E.t<.{t}, <.{y}, r>.>. e. ( SI (1st |` (`'2nd"{x})) (x) S ) <-> E.t(t = <.y, x>. /\ t e. r))
21	2, 20	bitri 240	. . . . . 6 |- (<.{y}, r>. e. (( SI (1st |` (`'2nd"{x})) (x) S )"1c) <-> E.t(t = <.y, x>. /\ t e. r))
22	5, 13	opex 4588	. . . . . . . 8 |- <.y, x>. e. _V
23		eleq1 2413	. . . . . . . 8 |- (t = <.y, x>. -> (t e. r <-> <.y, x>. e. r))
24	22, 23	ceqsexv 2894	. . . . . . 7 |- (E.t(t = <.y, x>. /\ t e. r) <-> <.y, x>. e. r)
25		df-br 4640	. . . . . . 7 |- (yrx <-> <.y, x>. e. r)
26	24, 25	bitr4i 243	. . . . . 6 |- (E.t(t = <.y, x>. /\ t e. r) <-> yrx)
27	21, 26	bitri 240	. . . . 5 |- (<.{y}, r>. e. (( SI (1st |` (`'2nd"{x})) (x) S )"1c) <-> yrx)
28		eliniseg 5020	. . . . 5 |- (y e. (`'r"{x}) <-> yrx)
29	27, 28	bitr4i 243	. . . 4 |- (<.{y}, r>. e. (( SI (1st |` (`'2nd"{x})) (x) S )"1c) <-> y e. (`'r"{x}))
30	29	releqmpt 5808	. . 3 |- (((A ^m B) X. _V) i^i `' ∼ (( Ins3 S (+) Ins2 (( SI (1st |` (`'2nd"{x})) (x) S )"1c))"1c)) = (r e. (A ^m B) |-> (`'r"{x}))
31	1, 30	eqtr4i 2376	. 2 |- W = (((A ^m B) X. _V) i^i `' ∼ (( Ins3 S (+) Ins2 (( SI (1st |` (`'2nd"{x})) (x) S )"1c))"1c))
32		ovex 5551	. . 3 |- (A ^m B) e. _V
33		1stex 4739	. . . . . . 7 |- 1st e. _V
34		2ndex 5112	. . . . . . . . 9 |- 2nd e. _V
35	34	cnvex 5102	. . . . . . . 8 |- `'2nd e. _V
36		snex 4111	. . . . . . . 8 |- {x} e. _V
37	35, 36	imaex 4747	. . . . . . 7 |- (`'2nd"{x}) e. _V
38	33, 37	resex 5117	. . . . . 6 |- (1st |` (`'2nd"{x})) e. _V
39	38	siex 4753	. . . . 5 |- SI (1st |` (`'2nd"{x})) e. _V
40		ssetex 4744	. . . . 5 |- S e. _V
41	39, 40	txpex 5785	. . . 4 |- ( SI (1st |` (`'2nd"{x})) (x) S ) e. _V
42		1cex 4142	. . . 4 |- 1c e. _V
43	41, 42	imaex 4747	. . 3 |- (( SI (1st |` (`'2nd"{x})) (x) S )"1c) e. _V
44	32, 43	mptexlem 5810	. 2 |- (((A ^m B) X. _V) i^i `' ∼ (( Ins3 S (+) Ins2 (( SI (1st |` (`'2nd"{x})) (x) S )"1c))"1c)) e. _V
45	31, 44	eqeltri 2423	1 |- W e. _V

Syntax hints:   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2859   ∼ ccompl 3205   i^i cin 3208   (+) csymdif 3209  {csn 3737  1_cc1c 4134  <.cop 4561   class class class wbr 4639  1stc1st 4717   S csset 4719   SI csi 4720  "cima 4722   X. cxp 4770  `'ccnv 4771   |` cres 4774  2ndc2nd 4783  (class class class)co 5525   |-> cmpt 5651   (x) ctxp 5735   Ins2 cins2 5749   Ins3 cins3 5751   ^m cmap 5999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fv 4795  df-2nd 4797  df-ov 5526  df-mpt 5652  df-txp 5736  df-ins2 5750  df-ins3 5752

This theorem is referenced by:  enprmap  6082