Theorem enmap1lem1 6070

Description: Lemma for enmap1 6075. Set up stratification. (Contributed by SF, 3-Mar-2015.)

Hypothesis

Ref	Expression
enmap1lem1.1	|- W = (s e. (A ^m G) |-> (r o. s))

Assertion

Ref	Expression
enmap1lem1	|- W e. _V

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

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

Proof of Theorem enmap1lem1

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

Step	Hyp	Ref	Expression
1		df-mpt 5653	. . 3 |- (s e. (A ^m G) |-> (r o. s)) = {<.s, x>. | (s e. (A ^m G) /\ x = (r o. s))}
2		enmap1lem1.1	. . 3 |- W = (s e. (A ^m G) |-> (r o. s))
3		opelres 4951	. . . . 5 |- (<.s, x>. e. (((((((`'1st"{r}) X. _V) i^i 2nd) o. 1st) (x) 2nd)" Compose ) |` (A ^m G)) <-> (<.s, x>. e. ((((((`'1st"{r}) X. _V) i^i 2nd) o. 1st) (x) 2nd)" Compose ) /\ s e. (A ^m G)))
4		elima 4755	. . . . . . . . 9 |- (<.s, x>. e. ((((((`'1st"{r}) X. _V) i^i 2nd) o. 1st) (x) 2nd)" Compose ) <-> E.p e. Compose p(((((`'1st"{r}) X. _V) i^i 2nd) o. 1st) (x) 2nd)<.s, x>.)
5		trtxp 5782	. . . . . . . . . . . 12 |- (p(((((`'1st"{r}) X. _V) i^i 2nd) o. 1st) (x) 2nd)<.s, x>. <-> (p((((`'1st"{r}) X. _V) i^i 2nd) o. 1st)s /\ p2ndx))
6		brco 4884	. . . . . . . . . . . . . 14 |- (p((((`'1st"{r}) X. _V) i^i 2nd) o. 1st)s <-> E.x(p1stx /\ x(((`'1st"{r}) X. _V) i^i 2nd)s))
7		ancom 437	. . . . . . . . . . . . . . . 16 |- ((p1stx /\ x(((`'1st"{r}) X. _V) i^i 2nd)s) <-> (x(((`'1st"{r}) X. _V) i^i 2nd)s /\ p1stx))
8		brin 4694	. . . . . . . . . . . . . . . . . 18 |- (x(((`'1st"{r}) X. _V) i^i 2nd)s <-> (x((`'1st"{r}) X. _V)s /\ x2nds))
9		vex 2863	. . . . . . . . . . . . . . . . . . . . 21 |- s e. _V
10		brxp 4813	. . . . . . . . . . . . . . . . . . . . 21 |- (x((`'1st"{r}) X. _V)s <-> (x e. (`'1st"{r}) /\ s e. _V))
11	9, 10	mpbiran2 885	. . . . . . . . . . . . . . . . . . . 20 |- (x((`'1st"{r}) X. _V)s <-> x e. (`'1st"{r}))
12		eliniseg 5021	. . . . . . . . . . . . . . . . . . . 20 |- (x e. (`'1st"{r}) <-> x1str)
13	11, 12	bitri 240	. . . . . . . . . . . . . . . . . . 19 |- (x((`'1st"{r}) X. _V)s <-> x1str)
14	13	anbi1i 676	. . . . . . . . . . . . . . . . . 18 |- ((x((`'1st"{r}) X. _V)s /\ x2nds) <-> (x1str /\ x2nds))
15		vex 2863	. . . . . . . . . . . . . . . . . . 19 |- r e. _V
16	15, 9	op1st2nd 5791	. . . . . . . . . . . . . . . . . 18 |- ((x1str /\ x2nds) <-> x = <.r, s>.)
17	8, 14, 16	3bitri 262	. . . . . . . . . . . . . . . . 17 |- (x(((`'1st"{r}) X. _V) i^i 2nd)s <-> x = <.r, s>.)
18	17	anbi1i 676	. . . . . . . . . . . . . . . 16 |- ((x(((`'1st"{r}) X. _V) i^i 2nd)s /\ p1stx) <-> (x = <.r, s>. /\ p1stx))
19	7, 18	bitri 240	. . . . . . . . . . . . . . 15 |- ((p1stx /\ x(((`'1st"{r}) X. _V) i^i 2nd)s) <-> (x = <.r, s>. /\ p1stx))
20	19	exbii 1582	. . . . . . . . . . . . . 14 |- (E.x(p1stx /\ x(((`'1st"{r}) X. _V) i^i 2nd)s) <-> E.x(x = <.r, s>. /\ p1stx))
21	15, 9	opex 4589	. . . . . . . . . . . . . . 15 |- <.r, s>. e. _V
22		breq2 4644	. . . . . . . . . . . . . . 15 |- (x = <.r, s>. -> (p1stx <-> p1st<.r, s>.))
23	21, 22	ceqsexv 2895	. . . . . . . . . . . . . 14 |- (E.x(x = <.r, s>. /\ p1stx) <-> p1st<.r, s>.)
24	6, 20, 23	3bitri 262	. . . . . . . . . . . . 13 |- (p((((`'1st"{r}) X. _V) i^i 2nd) o. 1st)s <-> p1st<.r, s>.)
25	24	anbi1i 676	. . . . . . . . . . . 12 |- ((p((((`'1st"{r}) X. _V) i^i 2nd) o. 1st)s /\ p2ndx) <-> (p1st<.r, s>. /\ p2ndx))
26		vex 2863	. . . . . . . . . . . . 13 |- x e. _V
27	21, 26	op1st2nd 5791	. . . . . . . . . . . 12 |- ((p1st<.r, s>. /\ p2ndx) <-> p = <.<.r, s>., x>.)
28	5, 25, 27	3bitri 262	. . . . . . . . . . 11 |- (p(((((`'1st"{r}) X. _V) i^i 2nd) o. 1st) (x) 2nd)<.s, x>. <-> p = <.<.r, s>., x>.)
29	28	rexbii 2640	. . . . . . . . . 10 |- (E.p e. Compose p(((((`'1st"{r}) X. _V) i^i 2nd) o. 1st) (x) 2nd)<.s, x>. <-> E.p e. Compose p = <.<.r, s>., x>.)
30		risset 2662	. . . . . . . . . 10 |- (<.<.r, s>., x>. e. Compose <-> E.p e. Compose p = <.<.r, s>., x>.)
31	29, 30	bitr4i 243	. . . . . . . . 9 |- (E.p e. Compose p(((((`'1st"{r}) X. _V) i^i 2nd) o. 1st) (x) 2nd)<.s, x>. <-> <.<.r, s>., x>. e. Compose )
32	4, 31	bitri 240	. . . . . . . 8 |- (<.s, x>. e. ((((((`'1st"{r}) X. _V) i^i 2nd) o. 1st) (x) 2nd)" Compose ) <-> <.<.r, s>., x>. e. Compose )
33		df-br 4641	. . . . . . . 8 |- (<.r, s>. Compose x <-> <.<.r, s>., x>. e. Compose )
34	32, 33	bitr4i 243	. . . . . . 7 |- (<.s, x>. e. ((((((`'1st"{r}) X. _V) i^i 2nd) o. 1st) (x) 2nd)" Compose ) <-> <.r, s>. Compose x)
35		brcomposeg 5820	. . . . . . . 8 |- ((r e. _V /\ s e. _V) -> (<.r, s>. Compose x <-> (r o. s) = x))
36	15, 9, 35	mp2an 653	. . . . . . 7 |- (<.r, s>. Compose x <-> (r o. s) = x)
37		eqcom 2355	. . . . . . 7 |- ((r o. s) = x <-> x = (r o. s))
38	34, 36, 37	3bitri 262	. . . . . 6 |- (<.s, x>. e. ((((((`'1st"{r}) X. _V) i^i 2nd) o. 1st) (x) 2nd)" Compose ) <-> x = (r o. s))
39	38	anbi2ci 677	. . . . 5 |- ((<.s, x>. e. ((((((`'1st"{r}) X. _V) i^i 2nd) o. 1st) (x) 2nd)" Compose ) /\ s e. (A ^m G)) <-> (s e. (A ^m G) /\ x = (r o. s)))
40	3, 39	bitri 240	. . . 4 |- (<.s, x>. e. (((((((`'1st"{r}) X. _V) i^i 2nd) o. 1st) (x) 2nd)" Compose ) |` (A ^m G)) <-> (s e. (A ^m G) /\ x = (r o. s)))
41	40	opabbi2i 4867	. . 3 |- (((((((`'1st"{r}) X. _V) i^i 2nd) o. 1st) (x) 2nd)" Compose ) |` (A ^m G)) = {<.s, x>. | (s e. (A ^m G) /\ x = (r o. s))}
42	1, 2, 41	3eqtr4i 2383	. 2 |- W = (((((((`'1st"{r}) X. _V) i^i 2nd) o. 1st) (x) 2nd)" Compose ) |` (A ^m G))
43		1stex 4740	. . . . . . . . . 10 |- 1st e. _V
44	43	cnvex 5103	. . . . . . . . 9 |- `'1st e. _V
45		snex 4112	. . . . . . . . 9 |- {r} e. _V
46	44, 45	imaex 4748	. . . . . . . 8 |- (`'1st"{r}) e. _V
47		vvex 4110	. . . . . . . 8 |- _V e. _V
48	46, 47	xpex 5116	. . . . . . 7 |- ((`'1st"{r}) X. _V) e. _V
49		2ndex 5113	. . . . . . 7 |- 2nd e. _V
50	48, 49	inex 4106	. . . . . 6 |- (((`'1st"{r}) X. _V) i^i 2nd) e. _V
51	50, 43	coex 4751	. . . . 5 |- ((((`'1st"{r}) X. _V) i^i 2nd) o. 1st) e. _V
52	51, 49	txpex 5786	. . . 4 |- (((((`'1st"{r}) X. _V) i^i 2nd) o. 1st) (x) 2nd) e. _V
53		composeex 5821	. . . 4 |- Compose e. _V
54	52, 53	imaex 4748	. . 3 |- ((((((`'1st"{r}) X. _V) i^i 2nd) o. 1st) (x) 2nd)" Compose ) e. _V
55		ovex 5552	. . 3 |- (A ^m G) e. _V
56	54, 55	resex 5118	. 2 |- (((((((`'1st"{r}) X. _V) i^i 2nd) o. 1st) (x) 2nd)" Compose ) |` (A ^m G)) 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 2616  _Vcvv 2860   i^i cin 3209  {csn 3738  <.cop 4562  {copab 4623   class class class wbr 4640  1stc1st 4718   o. ccom 4722  "cima 4723   X. cxp 4771  `'ccnv 4772   |` cres 4775  2ndc2nd 4784  (class class class)co 5526   |-> cmpt 5652   (x) ctxp 5736   Compose ccompose 5748   ^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-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-compose 5749  df-ins2 5751  df-ins3 5753  df-ins4 5757  df-si3 5759

This theorem is referenced by:  enmap1  6075