Theorem enprmaplem1 6077

Description: Lemma for enprmap 6083. 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 4948	. . . . . . 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 5783	. . . . . . . . 9 |- (<.{t}, <.{y}, r>.>. e. ( SI (1st |` (`'2nd"{x})) (x) S ) <-> (<.{t}, {y}>. e. SI (1st |` (`'2nd"{x})) /\ <.{t}, r>. e. S ))
4		vex 2863	. . . . . . . . . . . 12 |- t e. _V
5		vex 2863	. . . . . . . . . . . 12 |- y e. _V
6	4, 5	opsnelsi 5775	. . . . . . . . . . 11 |- (<.{t}, {y}>. e. SI (1st |` (`'2nd"{x})) <-> <.t, y>. e. (1st |` (`'2nd"{x})))
7		df-br 4641	. . . . . . . . . . . 12 |- (t(1st |` (`'2nd"{x}))y <-> <.t, y>. e. (1st |` (`'2nd"{x})))
8		brres 4950	. . . . . . . . . . . . 13 |- (t(1st |` (`'2nd"{x}))y <-> (t1sty /\ t e. (`'2nd"{x})))
9		eliniseg 5021	. . . . . . . . . . . . . 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 2863	. . . . . . . . . . . 12 |- x e. _V
14	5, 13	op1st2nd 5791	. . . . . . . . . . 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 2863	. . . . . . . . . . 11 |- r e. _V
17	4, 16	opelssetsn 4761	. . . . . . . . . 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 4589	. . . . . . . 8 |- <.y, x>. e. _V
23		eleq1 2413	. . . . . . . 8 |- (t = <.y, x>. -> (t e. r <-> <.y, x>. e. r))
24	22, 23	ceqsexv 2895	. . . . . . 7 |- (E.t(t = <.y, x>. /\ t e. r) <-> <.y, x>. e. r)
25		df-br 4641	. . . . . . 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 5021	. . . . 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 5809	. . 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 5552	. . 3 |- (A ^m B) e. _V
33		1stex 4740	. . . . . . 7 |- 1st e. _V
34		2ndex 5113	. . . . . . . . 9 |- 2nd e. _V
35	34	cnvex 5103	. . . . . . . 8 |- `'2nd e. _V
36		snex 4112	. . . . . . . 8 |- {x} e. _V
37	35, 36	imaex 4748	. . . . . . 7 |- (`'2nd"{x}) e. _V
38	33, 37	resex 5118	. . . . . 6 |- (1st |` (`'2nd"{x})) e. _V
39	38	siex 4754	. . . . 5 |- SI (1st |` (`'2nd"{x})) e. _V
40		ssetex 4745	. . . . 5 |- S e. _V
41	39, 40	txpex 5786	. . . 4 |- ( SI (1st |` (`'2nd"{x})) (x) S ) e. _V
42		1cex 4143	. . . 4 |- 1c e. _V
43	41, 42	imaex 4748	. . 3 |- (( SI (1st |` (`'2nd"{x})) (x) S )"1c) e. _V
44	32, 43	mptexlem 5811	. 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 2860   ∼ ccompl 3206   i^i cin 3209   (+) csymdif 3210  {csn 3738  1_cc1c 4135  <.cop 4562   class class class wbr 4640  1stc1st 4718   S csset 4720   SI csi 4721  "cima 4723   X. cxp 4771  `'ccnv 4772   |` cres 4775  2ndc2nd 4784  (class class class)co 5526   |-> cmpt 5652   (x) ctxp 5736   Ins2 cins2 5750   Ins3 cins3 5752   ^m cmap 6000

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fv 4796  df-2nd 4798  df-ov 5527  df-mpt 5653  df-txp 5737  df-ins2 5751  df-ins3 5753

This theorem is referenced by:  enprmap  6083