Theorem enmap1lem1 6070

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

Hypothesis

Ref	Expression
enmap1lem1.1	⊢ W = (s ∈ (A ↑m G) ↦ (r ∘ s))

Assertion

Ref	Expression
enmap1lem1	⊢ W ∈ 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 ∈ (A ↑m G) ↦ (r ∘ s)) = {⟨s, x⟩ ∣ (s ∈ (A ↑m G) ∧ x = (r ∘ s))}
2		enmap1lem1.1	. . 3 ⊢ W = (s ∈ (A ↑m G) ↦ (r ∘ s))
3		opelres 4951	. . . . 5 ⊢ (⟨s, x⟩ ∈ (((((((◡1st “ {r}) × V) ∩ 2nd ) ∘ 1st ) ⊗ 2nd ) “ Compose ) ↾ (A ↑m G)) ↔ (⟨s, x⟩ ∈ ((((((◡1st “ {r}) × V) ∩ 2nd ) ∘ 1st ) ⊗ 2nd ) “ Compose ) ∧ s ∈ (A ↑m G)))
4		elima 4755	. . . . . . . . 9 ⊢ (⟨s, x⟩ ∈ ((((((◡1st “ {r}) × V) ∩ 2nd ) ∘ 1st ) ⊗ 2nd ) “ Compose ) ↔ ∃p ∈ Compose p(((((◡1st “ {r}) × V) ∩ 2nd ) ∘ 1st ) ⊗ 2nd )⟨s, x⟩)
5		trtxp 5782	. . . . . . . . . . . 12 ⊢ (p(((((◡1st “ {r}) × V) ∩ 2nd ) ∘ 1st ) ⊗ 2nd )⟨s, x⟩ ↔ (p((((◡1st “ {r}) × V) ∩ 2nd ) ∘ 1st )s ∧ p2nd x))
6		brco 4884	. . . . . . . . . . . . . 14 ⊢ (p((((◡1st “ {r}) × V) ∩ 2nd ) ∘ 1st )s ↔ ∃x(p1st x ∧ x(((◡1st “ {r}) × V) ∩ 2nd )s))
7		ancom 437	. . . . . . . . . . . . . . . 16 ⊢ ((p1st x ∧ x(((◡1st “ {r}) × V) ∩ 2nd )s) ↔ (x(((◡1st “ {r}) × V) ∩ 2nd )s ∧ p1st x))
8		brin 4694	. . . . . . . . . . . . . . . . . 18 ⊢ (x(((◡1st “ {r}) × V) ∩ 2nd )s ↔ (x((◡1st “ {r}) × V)s ∧ x2nd s))
9		vex 2863	. . . . . . . . . . . . . . . . . . . . 21 ⊢ s ∈ V
10		brxp 4813	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (x((◡1st “ {r}) × V)s ↔ (x ∈ (◡1st “ {r}) ∧ s ∈ V))
11	9, 10	mpbiran2 885	. . . . . . . . . . . . . . . . . . . 20 ⊢ (x((◡1st “ {r}) × V)s ↔ x ∈ (◡1st “ {r}))
12		eliniseg 5021	. . . . . . . . . . . . . . . . . . . 20 ⊢ (x ∈ (◡1st “ {r}) ↔ x1st r)
13	11, 12	bitri 240	. . . . . . . . . . . . . . . . . . 19 ⊢ (x((◡1st “ {r}) × V)s ↔ x1st r)
14	13	anbi1i 676	. . . . . . . . . . . . . . . . . 18 ⊢ ((x((◡1st “ {r}) × V)s ∧ x2nd s) ↔ (x1st r ∧ x2nd s))
15		vex 2863	. . . . . . . . . . . . . . . . . . 19 ⊢ r ∈ V
16	15, 9	op1st2nd 5791	. . . . . . . . . . . . . . . . . 18 ⊢ ((x1st r ∧ x2nd s) ↔ x = ⟨r, s⟩)
17	8, 14, 16	3bitri 262	. . . . . . . . . . . . . . . . 17 ⊢ (x(((◡1st “ {r}) × V) ∩ 2nd )s ↔ x = ⟨r, s⟩)
18	17	anbi1i 676	. . . . . . . . . . . . . . . 16 ⊢ ((x(((◡1st “ {r}) × V) ∩ 2nd )s ∧ p1st x) ↔ (x = ⟨r, s⟩ ∧ p1st x))
19	7, 18	bitri 240	. . . . . . . . . . . . . . 15 ⊢ ((p1st x ∧ x(((◡1st “ {r}) × V) ∩ 2nd )s) ↔ (x = ⟨r, s⟩ ∧ p1st x))
20	19	exbii 1582	. . . . . . . . . . . . . 14 ⊢ (∃x(p1st x ∧ x(((◡1st “ {r}) × V) ∩ 2nd )s) ↔ ∃x(x = ⟨r, s⟩ ∧ p1st x))
21	15, 9	opex 4589	. . . . . . . . . . . . . . 15 ⊢ ⟨r, s⟩ ∈ V
22		breq2 4644	. . . . . . . . . . . . . . 15 ⊢ (x = ⟨r, s⟩ → (p1st x ↔ p1st ⟨r, s⟩))
23	21, 22	ceqsexv 2895	. . . . . . . . . . . . . 14 ⊢ (∃x(x = ⟨r, s⟩ ∧ p1st x) ↔ p1st ⟨r, s⟩)
24	6, 20, 23	3bitri 262	. . . . . . . . . . . . 13 ⊢ (p((((◡1st “ {r}) × V) ∩ 2nd ) ∘ 1st )s ↔ p1st ⟨r, s⟩)
25	24	anbi1i 676	. . . . . . . . . . . 12 ⊢ ((p((((◡1st “ {r}) × V) ∩ 2nd ) ∘ 1st )s ∧ p2nd x) ↔ (p1st ⟨r, s⟩ ∧ p2nd x))
26		vex 2863	. . . . . . . . . . . . 13 ⊢ x ∈ V
27	21, 26	op1st2nd 5791	. . . . . . . . . . . 12 ⊢ ((p1st ⟨r, s⟩ ∧ p2nd x) ↔ p = ⟨⟨r, s⟩, x⟩)
28	5, 25, 27	3bitri 262	. . . . . . . . . . 11 ⊢ (p(((((◡1st “ {r}) × V) ∩ 2nd ) ∘ 1st ) ⊗ 2nd )⟨s, x⟩ ↔ p = ⟨⟨r, s⟩, x⟩)
29	28	rexbii 2640	. . . . . . . . . 10 ⊢ (∃p ∈ Compose p(((((◡1st “ {r}) × V) ∩ 2nd ) ∘ 1st ) ⊗ 2nd )⟨s, x⟩ ↔ ∃p ∈ Compose p = ⟨⟨r, s⟩, x⟩)
30		risset 2662	. . . . . . . . . 10 ⊢ (⟨⟨r, s⟩, x⟩ ∈ Compose ↔ ∃p ∈ Compose p = ⟨⟨r, s⟩, x⟩)
31	29, 30	bitr4i 243	. . . . . . . . 9 ⊢ (∃p ∈ Compose p(((((◡1st “ {r}) × V) ∩ 2nd ) ∘ 1st ) ⊗ 2nd )⟨s, x⟩ ↔ ⟨⟨r, s⟩, x⟩ ∈ Compose )
32	4, 31	bitri 240	. . . . . . . 8 ⊢ (⟨s, x⟩ ∈ ((((((◡1st “ {r}) × V) ∩ 2nd ) ∘ 1st ) ⊗ 2nd ) “ Compose ) ↔ ⟨⟨r, s⟩, x⟩ ∈ Compose )
33		df-br 4641	. . . . . . . 8 ⊢ (⟨r, s⟩ Compose x ↔ ⟨⟨r, s⟩, x⟩ ∈ Compose )
34	32, 33	bitr4i 243	. . . . . . 7 ⊢ (⟨s, x⟩ ∈ ((((((◡1st “ {r}) × V) ∩ 2nd ) ∘ 1st ) ⊗ 2nd ) “ Compose ) ↔ ⟨r, s⟩ Compose x)
35		brcomposeg 5820	. . . . . . . 8 ⊢ ((r ∈ V ∧ s ∈ V) → (⟨r, s⟩ Compose x ↔ (r ∘ s) = x))
36	15, 9, 35	mp2an 653	. . . . . . 7 ⊢ (⟨r, s⟩ Compose x ↔ (r ∘ s) = x)
37		eqcom 2355	. . . . . . 7 ⊢ ((r ∘ s) = x ↔ x = (r ∘ s))
38	34, 36, 37	3bitri 262	. . . . . 6 ⊢ (⟨s, x⟩ ∈ ((((((◡1st “ {r}) × V) ∩ 2nd ) ∘ 1st ) ⊗ 2nd ) “ Compose ) ↔ x = (r ∘ s))
39	38	anbi2ci 677	. . . . 5 ⊢ ((⟨s, x⟩ ∈ ((((((◡1st “ {r}) × V) ∩ 2nd ) ∘ 1st ) ⊗ 2nd ) “ Compose ) ∧ s ∈ (A ↑m G)) ↔ (s ∈ (A ↑m G) ∧ x = (r ∘ s)))
40	3, 39	bitri 240	. . . 4 ⊢ (⟨s, x⟩ ∈ (((((((◡1st “ {r}) × V) ∩ 2nd ) ∘ 1st ) ⊗ 2nd ) “ Compose ) ↾ (A ↑m G)) ↔ (s ∈ (A ↑m G) ∧ x = (r ∘ s)))
41	40	opabbi2i 4867	. . 3 ⊢ (((((((◡1st “ {r}) × V) ∩ 2nd ) ∘ 1st ) ⊗ 2nd ) “ Compose ) ↾ (A ↑m G)) = {⟨s, x⟩ ∣ (s ∈ (A ↑m G) ∧ x = (r ∘ s))}
42	1, 2, 41	3eqtr4i 2383	. 2 ⊢ W = (((((((◡1st “ {r}) × V) ∩ 2nd ) ∘ 1st ) ⊗ 2nd ) “ Compose ) ↾ (A ↑m G))
43		1stex 4740	. . . . . . . . . 10 ⊢ 1st ∈ V
44	43	cnvex 5103	. . . . . . . . 9 ⊢ ◡1st ∈ V
45		snex 4112	. . . . . . . . 9 ⊢ {r} ∈ V
46	44, 45	imaex 4748	. . . . . . . 8 ⊢ (◡1st “ {r}) ∈ V
47		vvex 4110	. . . . . . . 8 ⊢ V ∈ V
48	46, 47	xpex 5116	. . . . . . 7 ⊢ ((◡1st “ {r}) × V) ∈ V
49		2ndex 5113	. . . . . . 7 ⊢ 2nd ∈ V
50	48, 49	inex 4106	. . . . . 6 ⊢ (((◡1st “ {r}) × V) ∩ 2nd ) ∈ V
51	50, 43	coex 4751	. . . . 5 ⊢ ((((◡1st “ {r}) × V) ∩ 2nd ) ∘ 1st ) ∈ V
52	51, 49	txpex 5786	. . . 4 ⊢ (((((◡1st “ {r}) × V) ∩ 2nd ) ∘ 1st ) ⊗ 2nd ) ∈ V
53		composeex 5821	. . . 4 ⊢ Compose ∈ V
54	52, 53	imaex 4748	. . 3 ⊢ ((((((◡1st “ {r}) × V) ∩ 2nd ) ∘ 1st ) ⊗ 2nd ) “ Compose ) ∈ V
55		ovex 5552	. . 3 ⊢ (A ↑m G) ∈ V
56	54, 55	resex 5118	. 2 ⊢ (((((((◡1st “ {r}) × V) ∩ 2nd ) ∘ 1st ) ⊗ 2nd ) “ Compose ) ↾ (A ↑m G)) ∈ V
57	42, 56	eqeltri 2423	1 ⊢ W ∈ V

This proof depends on syntax axioms:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  Vcvv 2860   ∩ cin 3209  {csn 3738  ⟨cop 4562  {copab 4623   class class class wbr 4640  1^st c1st 4718   ∘ ccom 4722   “ cima 4723   × cxp 4771  ^◡ccnv 4772   ↾ cres 4775  2^nd c2nd 4784  (class class class)co 5526   ↦ cmpt 5652   ⊗ ctxp 5736   Compose ccompose 5748   ↑_m cmap 6000

This proof depends on 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 proof 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 used by:  enmap1  6075