Theorem ceexlem1 6174

Description: Lemma for ceex 6175. Set up part of the stratification. (Contributed by SF, 6-Mar-2015.)

Assertion

Ref	Expression
ceexlem1	⊢ (⟨{{a}}, n⟩ ∈ ( S ∘ SI Pw1Fn ) ↔ ℘1a ∈ n)

Distinct variable group:   n,a

Proof of Theorem ceexlem1

Dummy variables t u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snex 4112	. . . . . . . 8 ⊢ {a} ∈ V
2	1	brsnsi1 5776	. . . . . . 7 ⊢ ({{a}} SI Pw1Fn u ↔ ∃t(u = {t} ∧ {a} Pw1Fn t))
3	2	anbi1i 676	. . . . . 6 ⊢ (({{a}} SI Pw1Fn u ∧ u S n) ↔ (∃t(u = {t} ∧ {a} Pw1Fn t) ∧ u S n))
4		19.41v 1901	. . . . . 6 ⊢ (∃t((u = {t} ∧ {a} Pw1Fn t) ∧ u S n) ↔ (∃t(u = {t} ∧ {a} Pw1Fn t) ∧ u S n))
5		anass 630	. . . . . . 7 ⊢ (((u = {t} ∧ {a} Pw1Fn t) ∧ u S n) ↔ (u = {t} ∧ ({a} Pw1Fn t ∧ u S n)))
6	5	exbii 1582	. . . . . 6 ⊢ (∃t((u = {t} ∧ {a} Pw1Fn t) ∧ u S n) ↔ ∃t(u = {t} ∧ ({a} Pw1Fn t ∧ u S n)))
7	3, 4, 6	3bitr2i 264	. . . . 5 ⊢ (({{a}} SI Pw1Fn u ∧ u S n) ↔ ∃t(u = {t} ∧ ({a} Pw1Fn t ∧ u S n)))
8	7	exbii 1582	. . . 4 ⊢ (∃u({{a}} SI Pw1Fn u ∧ u S n) ↔ ∃u∃t(u = {t} ∧ ({a} Pw1Fn t ∧ u S n)))
9		excom 1741	. . . 4 ⊢ (∃u∃t(u = {t} ∧ ({a} Pw1Fn t ∧ u S n)) ↔ ∃t∃u(u = {t} ∧ ({a} Pw1Fn t ∧ u S n)))
10	8, 9	bitri 240	. . 3 ⊢ (∃u({{a}} SI Pw1Fn u ∧ u S n) ↔ ∃t∃u(u = {t} ∧ ({a} Pw1Fn t ∧ u S n)))
11		snex 4112	. . . . . 6 ⊢ {t} ∈ V
12		breq1 4643	. . . . . . 7 ⊢ (u = {t} → (u S n ↔ {t} S n))
13	12	anbi2d 684	. . . . . 6 ⊢ (u = {t} → (({a} Pw1Fn t ∧ u S n) ↔ ({a} Pw1Fn t ∧ {t} S n)))
14	11, 13	ceqsexv 2895	. . . . 5 ⊢ (∃u(u = {t} ∧ ({a} Pw1Fn t ∧ u S n)) ↔ ({a} Pw1Fn t ∧ {t} S n))
15		vex 2863	. . . . . . 7 ⊢ a ∈ V
16	15	brpw1fn 5855	. . . . . 6 ⊢ ({a} Pw1Fn t ↔ t = ℘1a)
17		vex 2863	. . . . . . 7 ⊢ t ∈ V
18		vex 2863	. . . . . . 7 ⊢ n ∈ V
19	17, 18	brssetsn 4760	. . . . . 6 ⊢ ({t} S n ↔ t ∈ n)
20	16, 19	anbi12i 678	. . . . 5 ⊢ (({a} Pw1Fn t ∧ {t} S n) ↔ (t = ℘1a ∧ t ∈ n))
21	14, 20	bitri 240	. . . 4 ⊢ (∃u(u = {t} ∧ ({a} Pw1Fn t ∧ u S n)) ↔ (t = ℘1a ∧ t ∈ n))
22	21	exbii 1582	. . 3 ⊢ (∃t∃u(u = {t} ∧ ({a} Pw1Fn t ∧ u S n)) ↔ ∃t(t = ℘1a ∧ t ∈ n))
23	10, 22	bitri 240	. 2 ⊢ (∃u({{a}} SI Pw1Fn u ∧ u S n) ↔ ∃t(t = ℘1a ∧ t ∈ n))
24		opelco 4885	. 2 ⊢ (⟨{{a}}, n⟩ ∈ ( S ∘ SI Pw1Fn ) ↔ ∃u({{a}} SI Pw1Fn u ∧ u S n))
25		df-clel 2349	. 2 ⊢ (℘1a ∈ n ↔ ∃t(t = ℘1a ∧ t ∈ n))
26	23, 24, 25	3bitr4i 268	1 ⊢ (⟨{{a}}, n⟩ ∈ ( S ∘ SI Pw1Fn ) ↔ ℘1a ∈ n)

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {csn 3738  ℘₁cpw1 4136  ⟨cop 4562   class class class wbr 4640   S csset 4720   SI csi 4721   ∘ ccom 4722   Pw1Fn cpw1fn 5766

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-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-fv 4796  df-mpt 5653  df-pw1fn 5767

This theorem is referenced by:  ceex  6175