Theorem enmap2lem1 6063

Description: Lemma for enmap2 6068. Set up stratification. (Contributed by SF, 26-Feb-2015.)

Hypothesis

Ref	Expression
enmap2lem1.1	|- W = (s e. (G ^m A) |-> (s o. `'r))

Assertion

Ref	Expression
enmap2lem1	|- W e. _V

Distinct variable groups:   A,s   G,s   s,r

Allowed substitution hints:   A(r)   G(r)   W(s,r)

Proof of Theorem enmap2lem1

Dummy variables p x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpt 5652	. . 3 |- (s e. (G ^m A) |-> (s o. `'r)) = {<.s, x>. | (s e. (G ^m A) /\ x = (s o. `'r))}
2		enmap2lem1.1	. . 3 |- W = (s e. (G ^m A) |-> (s o. `'r))
3		opelres 4950	. . . . 5 |- (<.s, x>. e. (((((1st i^i ((`'2nd"{`'r}) X. _V)) o. 1st) (x) 2nd)" Compose ) |` (G ^m A)) <-> (<.s, x>. e. ((((1st i^i ((`'2nd"{`'r}) X. _V)) o. 1st) (x) 2nd)" Compose ) /\ s e. (G ^m A)))
4		trtxp 5781	. . . . . . . . . 10 |- (p(((1st i^i ((`'2nd"{`'r}) X. _V)) o. 1st) (x) 2nd)<.s, x>. <-> (p((1st i^i ((`'2nd"{`'r}) X. _V)) o. 1st)s /\ p2ndx))
5		brco 4883	. . . . . . . . . . . 12 |- (p((1st i^i ((`'2nd"{`'r}) X. _V)) o. 1st)s <-> E.x(p1stx /\ x(1st i^i ((`'2nd"{`'r}) X. _V))s))
6		ancom 437	. . . . . . . . . . . . . 14 |- ((p1stx /\ x(1st i^i ((`'2nd"{`'r}) X. _V))s) <-> (x(1st i^i ((`'2nd"{`'r}) X. _V))s /\ p1stx))
7		brin 4693	. . . . . . . . . . . . . . . 16 |- (x(1st i^i ((`'2nd"{`'r}) X. _V))s <-> (x1sts /\ x((`'2nd"{`'r}) X. _V)s))
8		vex 2862	. . . . . . . . . . . . . . . . . . 19 |- s e. _V
9		brxp 4812	. . . . . . . . . . . . . . . . . . 19 |- (x((`'2nd"{`'r}) X. _V)s <-> (x e. (`'2nd"{`'r}) /\ s e. _V))
10	8, 9	mpbiran2 885	. . . . . . . . . . . . . . . . . 18 |- (x((`'2nd"{`'r}) X. _V)s <-> x e. (`'2nd"{`'r}))
11		eliniseg 5020	. . . . . . . . . . . . . . . . . 18 |- (x e. (`'2nd"{`'r}) <-> x2nd`'r)
12	10, 11	bitri 240	. . . . . . . . . . . . . . . . 17 |- (x((`'2nd"{`'r}) X. _V)s <-> x2nd`'r)
13	12	anbi2i 675	. . . . . . . . . . . . . . . 16 |- ((x1sts /\ x((`'2nd"{`'r}) X. _V)s) <-> (x1sts /\ x2nd`'r))
14		vex 2862	. . . . . . . . . . . . . . . . . 18 |- r e. _V
15	14	cnvex 5102	. . . . . . . . . . . . . . . . 17 |- `'r e. _V
16	8, 15	op1st2nd 5790	. . . . . . . . . . . . . . . 16 |- ((x1sts /\ x2nd`'r) <-> x = <.s, `'r>.)
17	7, 13, 16	3bitri 262	. . . . . . . . . . . . . . 15 |- (x(1st i^i ((`'2nd"{`'r}) X. _V))s <-> x = <.s, `'r>.)
18	17	anbi1i 676	. . . . . . . . . . . . . 14 |- ((x(1st i^i ((`'2nd"{`'r}) X. _V))s /\ p1stx) <-> (x = <.s, `'r>. /\ p1stx))
19	6, 18	bitri 240	. . . . . . . . . . . . 13 |- ((p1stx /\ x(1st i^i ((`'2nd"{`'r}) X. _V))s) <-> (x = <.s, `'r>. /\ p1stx))
20	19	exbii 1582	. . . . . . . . . . . 12 |- (E.x(p1stx /\ x(1st i^i ((`'2nd"{`'r}) X. _V))s) <-> E.x(x = <.s, `'r>. /\ p1stx))
21	8, 15	opex 4588	. . . . . . . . . . . . 13 |- <.s, `'r>. e. _V
22		breq2 4643	. . . . . . . . . . . . 13 |- (x = <.s, `'r>. -> (p1stx <-> p1st<.s, `'r>.))
23	21, 22	ceqsexv 2894	. . . . . . . . . . . 12 |- (E.x(x = <.s, `'r>. /\ p1stx) <-> p1st<.s, `'r>.)
24	5, 20, 23	3bitri 262	. . . . . . . . . . 11 |- (p((1st i^i ((`'2nd"{`'r}) X. _V)) o. 1st)s <-> p1st<.s, `'r>.)
25	24	anbi1i 676	. . . . . . . . . 10 |- ((p((1st i^i ((`'2nd"{`'r}) X. _V)) o. 1st)s /\ p2ndx) <-> (p1st<.s, `'r>. /\ p2ndx))
26		vex 2862	. . . . . . . . . . 11 |- x e. _V
27	21, 26	op1st2nd 5790	. . . . . . . . . 10 |- ((p1st<.s, `'r>. /\ p2ndx) <-> p = <.<.s, `'r>., x>.)
28	4, 25, 27	3bitri 262	. . . . . . . . 9 |- (p(((1st i^i ((`'2nd"{`'r}) X. _V)) o. 1st) (x) 2nd)<.s, x>. <-> p = <.<.s, `'r>., x>.)
29	28	rexbii 2639	. . . . . . . 8 |- (E.p e. Compose p(((1st i^i ((`'2nd"{`'r}) X. _V)) o. 1st) (x) 2nd)<.s, x>. <-> E.p e. Compose p = <.<.s, `'r>., x>.)
30		elima 4754	. . . . . . . 8 |- (<.s, x>. e. ((((1st i^i ((`'2nd"{`'r}) X. _V)) o. 1st) (x) 2nd)" Compose ) <-> E.p e. Compose p(((1st i^i ((`'2nd"{`'r}) X. _V)) o. 1st) (x) 2nd)<.s, x>.)
31		risset 2661	. . . . . . . 8 |- (<.<.s, `'r>., x>. e. Compose <-> E.p e. Compose p = <.<.s, `'r>., x>.)
32	29, 30, 31	3bitr4i 268	. . . . . . 7 |- (<.s, x>. e. ((((1st i^i ((`'2nd"{`'r}) X. _V)) o. 1st) (x) 2nd)" Compose ) <-> <.<.s, `'r>., x>. e. Compose )
33		df-br 4640	. . . . . . 7 |- (<.s, `'r>. Compose x <-> <.<.s, `'r>., x>. e. Compose )
34		brcomposeg 5819	. . . . . . . . 9 |- ((s e. _V /\ `'r e. _V) -> (<.s, `'r>. Compose x <-> (s o. `'r) = x))
35	8, 15, 34	mp2an 653	. . . . . . . 8 |- (<.s, `'r>. Compose x <-> (s o. `'r) = x)
36		eqcom 2355	. . . . . . . 8 |- ((s o. `'r) = x <-> x = (s o. `'r))
37	35, 36	bitri 240	. . . . . . 7 |- (<.s, `'r>. Compose x <-> x = (s o. `'r))
38	32, 33, 37	3bitr2i 264	. . . . . 6 |- (<.s, x>. e. ((((1st i^i ((`'2nd"{`'r}) X. _V)) o. 1st) (x) 2nd)" Compose ) <-> x = (s o. `'r))
39	38	anbi2ci 677	. . . . 5 |- ((<.s, x>. e. ((((1st i^i ((`'2nd"{`'r}) X. _V)) o. 1st) (x) 2nd)" Compose ) /\ s e. (G ^m A)) <-> (s e. (G ^m A) /\ x = (s o. `'r)))
40	3, 39	bitri 240	. . . 4 |- (<.s, x>. e. (((((1st i^i ((`'2nd"{`'r}) X. _V)) o. 1st) (x) 2nd)" Compose ) |` (G ^m A)) <-> (s e. (G ^m A) /\ x = (s o. `'r)))
41	40	opabbi2i 4866	. . 3 |- (((((1st i^i ((`'2nd"{`'r}) X. _V)) o. 1st) (x) 2nd)" Compose ) |` (G ^m A)) = {<.s, x>. | (s e. (G ^m A) /\ x = (s o. `'r))}
42	1, 2, 41	3eqtr4i 2383	. 2 |- W = (((((1st i^i ((`'2nd"{`'r}) X. _V)) o. 1st) (x) 2nd)" Compose ) |` (G ^m A))
43		1stex 4739	. . . . . . 7 |- 1st e. _V
44		2ndex 5112	. . . . . . . . . 10 |- 2nd e. _V
45	44	cnvex 5102	. . . . . . . . 9 |- `'2nd e. _V
46		snex 4111	. . . . . . . . 9 |- {`'r} e. _V
47	45, 46	imaex 4747	. . . . . . . 8 |- (`'2nd"{`'r}) e. _V
48		vvex 4109	. . . . . . . 8 |- _V e. _V
49	47, 48	xpex 5115	. . . . . . 7 |- ((`'2nd"{`'r}) X. _V) e. _V
50	43, 49	inex 4105	. . . . . 6 |- (1st i^i ((`'2nd"{`'r}) X. _V)) e. _V
51	50, 43	coex 4750	. . . . 5 |- ((1st i^i ((`'2nd"{`'r}) X. _V)) o. 1st) e. _V
52	51, 44	txpex 5785	. . . 4 |- (((1st i^i ((`'2nd"{`'r}) X. _V)) o. 1st) (x) 2nd) e. _V
53		composeex 5820	. . . 4 |- Compose e. _V
54	52, 53	imaex 4747	. . 3 |- ((((1st i^i ((`'2nd"{`'r}) X. _V)) o. 1st) (x) 2nd)" Compose ) e. _V
55		ovex 5551	. . 3 |- (G ^m A) e. _V
56	54, 55	resex 5117	. 2 |- (((((1st i^i ((`'2nd"{`'r}) X. _V)) o. 1st) (x) 2nd)" Compose ) |` (G ^m A)) e. _V
57	42, 56	eqeltri 2423	1 |- W e. _V

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  E.wrex 2615  _Vcvv 2859   i^i cin 3208  {csn 3737  <.cop 4561  {copab 4622   class class class wbr 4639  1stc1st 4717   o. ccom 4721  "cima 4722   X. cxp 4770  `'ccnv 4771   |` cres 4774  2ndc2nd 4783  (class class class)co 5525   |-> cmpt 5651   (x) ctxp 5735   Compose ccompose 5747   ^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-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-compose 5748  df-ins2 5750  df-ins3 5752  df-ins4 5756  df-si3 5758

This theorem is referenced by:  enmap2  6068