Theorem enprmaplem1 6077

Description: Lemma for enprmap 6083. Set up stratification. (Contributed by SF, 3-Mar-2015.)

Hypothesis

Ref	Expression
enprmaplem1.1	⊢ W = (r ∈ (A ↑m B) ↦ (◡r “ {x}))

Assertion

Ref	Expression
enprmaplem1	⊢ W ∈ 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 ∈ (A ↑m B) ↦ (◡r “ {x}))
2		elima1c 4948	. . . . . . 7 ⊢ (⟨{y}, r⟩ ∈ (( SI (1st ↾ (◡2nd “ {x})) ⊗ S ) “ 1c) ↔ ∃t⟨{t}, ⟨{y}, r⟩⟩ ∈ ( SI (1st ↾ (◡2nd “ {x})) ⊗ S ))
3		oteltxp 5783	. . . . . . . . 9 ⊢ (⟨{t}, ⟨{y}, r⟩⟩ ∈ ( SI (1st ↾ (◡2nd “ {x})) ⊗ S ) ↔ (⟨{t}, {y}⟩ ∈ SI (1st ↾ (◡2nd “ {x})) ∧ ⟨{t}, r⟩ ∈ S ))
4		vex 2863	. . . . . . . . . . . 12 ⊢ t ∈ V
5		vex 2863	. . . . . . . . . . . 12 ⊢ y ∈ V
6	4, 5	opsnelsi 5775	. . . . . . . . . . 11 ⊢ (⟨{t}, {y}⟩ ∈ SI (1st ↾ (◡2nd “ {x})) ↔ ⟨t, y⟩ ∈ (1st ↾ (◡2nd “ {x})))
7		df-br 4641	. . . . . . . . . . . 12 ⊢ (t(1st ↾ (◡2nd “ {x}))y ↔ ⟨t, y⟩ ∈ (1st ↾ (◡2nd “ {x})))
8		brres 4950	. . . . . . . . . . . . 13 ⊢ (t(1st ↾ (◡2nd “ {x}))y ↔ (t1st y ∧ t ∈ (◡2nd “ {x})))
9		eliniseg 5021	. . . . . . . . . . . . . 14 ⊢ (t ∈ (◡2nd “ {x}) ↔ t2nd x)
10	9	anbi2i 675	. . . . . . . . . . . . 13 ⊢ ((t1st y ∧ t ∈ (◡2nd “ {x})) ↔ (t1st y ∧ t2nd x))
11	8, 10	bitri 240	. . . . . . . . . . . 12 ⊢ (t(1st ↾ (◡2nd “ {x}))y ↔ (t1st y ∧ t2nd x))
12	7, 11	bitr3i 242	. . . . . . . . . . 11 ⊢ (⟨t, y⟩ ∈ (1st ↾ (◡2nd “ {x})) ↔ (t1st y ∧ t2nd x))
13		vex 2863	. . . . . . . . . . . 12 ⊢ x ∈ V
14	5, 13	op1st2nd 5791	. . . . . . . . . . 11 ⊢ ((t1st y ∧ t2nd x) ↔ t = ⟨y, x⟩)
15	6, 12, 14	3bitri 262	. . . . . . . . . 10 ⊢ (⟨{t}, {y}⟩ ∈ SI (1st ↾ (◡2nd “ {x})) ↔ t = ⟨y, x⟩)
16		vex 2863	. . . . . . . . . . 11 ⊢ r ∈ V
17	4, 16	opelssetsn 4761	. . . . . . . . . 10 ⊢ (⟨{t}, r⟩ ∈ S ↔ t ∈ r)
18	15, 17	anbi12i 678	. . . . . . . . 9 ⊢ ((⟨{t}, {y}⟩ ∈ SI (1st ↾ (◡2nd “ {x})) ∧ ⟨{t}, r⟩ ∈ S ) ↔ (t = ⟨y, x⟩ ∧ t ∈ r))
19	3, 18	bitri 240	. . . . . . . 8 ⊢ (⟨{t}, ⟨{y}, r⟩⟩ ∈ ( SI (1st ↾ (◡2nd “ {x})) ⊗ S ) ↔ (t = ⟨y, x⟩ ∧ t ∈ r))
20	19	exbii 1582	. . . . . . 7 ⊢ (∃t⟨{t}, ⟨{y}, r⟩⟩ ∈ ( SI (1st ↾ (◡2nd “ {x})) ⊗ S ) ↔ ∃t(t = ⟨y, x⟩ ∧ t ∈ r))
21	2, 20	bitri 240	. . . . . 6 ⊢ (⟨{y}, r⟩ ∈ (( SI (1st ↾ (◡2nd “ {x})) ⊗ S ) “ 1c) ↔ ∃t(t = ⟨y, x⟩ ∧ t ∈ r))
22	5, 13	opex 4589	. . . . . . . 8 ⊢ ⟨y, x⟩ ∈ V
23		eleq1 2413	. . . . . . . 8 ⊢ (t = ⟨y, x⟩ → (t ∈ r ↔ ⟨y, x⟩ ∈ r))
24	22, 23	ceqsexv 2895	. . . . . . 7 ⊢ (∃t(t = ⟨y, x⟩ ∧ t ∈ r) ↔ ⟨y, x⟩ ∈ r)
25		df-br 4641	. . . . . . 7 ⊢ (yrx ↔ ⟨y, x⟩ ∈ r)
26	24, 25	bitr4i 243	. . . . . 6 ⊢ (∃t(t = ⟨y, x⟩ ∧ t ∈ r) ↔ yrx)
27	21, 26	bitri 240	. . . . 5 ⊢ (⟨{y}, r⟩ ∈ (( SI (1st ↾ (◡2nd “ {x})) ⊗ S ) “ 1c) ↔ yrx)
28		eliniseg 5021	. . . . 5 ⊢ (y ∈ (◡r “ {x}) ↔ yrx)
29	27, 28	bitr4i 243	. . . 4 ⊢ (⟨{y}, r⟩ ∈ (( SI (1st ↾ (◡2nd “ {x})) ⊗ S ) “ 1c) ↔ y ∈ (◡r “ {x}))
30	29	releqmpt 5809	. . 3 ⊢ (((A ↑m B) × V) ∩ ◡ ∼ (( Ins3 S ⊕ Ins2 (( SI (1st ↾ (◡2nd “ {x})) ⊗ S ) “ 1c)) “ 1c)) = (r ∈ (A ↑m B) ↦ (◡r “ {x}))
31	1, 30	eqtr4i 2376	. 2 ⊢ W = (((A ↑m B) × V) ∩ ◡ ∼ (( Ins3 S ⊕ Ins2 (( SI (1st ↾ (◡2nd “ {x})) ⊗ S ) “ 1c)) “ 1c))
32		ovex 5552	. . 3 ⊢ (A ↑m B) ∈ V
33		1stex 4740	. . . . . . 7 ⊢ 1st ∈ V
34		2ndex 5113	. . . . . . . . 9 ⊢ 2nd ∈ V
35	34	cnvex 5103	. . . . . . . 8 ⊢ ◡2nd ∈ V
36		snex 4112	. . . . . . . 8 ⊢ {x} ∈ V
37	35, 36	imaex 4748	. . . . . . 7 ⊢ (◡2nd “ {x}) ∈ V
38	33, 37	resex 5118	. . . . . 6 ⊢ (1st ↾ (◡2nd “ {x})) ∈ V
39	38	siex 4754	. . . . 5 ⊢ SI (1st ↾ (◡2nd “ {x})) ∈ V
40		ssetex 4745	. . . . 5 ⊢ S ∈ V
41	39, 40	txpex 5786	. . . 4 ⊢ ( SI (1st ↾ (◡2nd “ {x})) ⊗ S ) ∈ V
42		1cex 4143	. . . 4 ⊢ 1c ∈ V
43	41, 42	imaex 4748	. . 3 ⊢ (( SI (1st ↾ (◡2nd “ {x})) ⊗ S ) “ 1c) ∈ V
44	32, 43	mptexlem 5811	. 2 ⊢ (((A ↑m B) × V) ∩ ◡ ∼ (( Ins3 S ⊕ Ins2 (( SI (1st ↾ (◡2nd “ {x})) ⊗ S ) “ 1c)) “ 1c)) ∈ V
45	31, 44	eqeltri 2423	1 ⊢ W ∈ V

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ∼ ccompl 3206   ∩ cin 3209   ⊕ csymdif 3210  {csn 3738  1_cc1c 4135  ⟨cop 4562   class class class wbr 4640  1^st c1st 4718   S csset 4720   SI csi 4721   “ cima 4723   × cxp 4771  ^◡ccnv 4772   ↾ cres 4775  2^nd c2nd 4784  (class class class)co 5526   ↦ cmpt 5652   ⊗ 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