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Theorem ceexlem1 6173
 Description: Lemma for ceex 6174. 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.
StepHypRef Expression
1 snex 4111 . . . . . . . 8 {a} V
21brsnsi1 5775 . . . . . . 7 ({{a}} SI Pw1Fn ut(u = {t} {a} Pw1Fn t))
32anbi1i 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)))
65exbii 1582 . . . . . 6 (t((u = {t} {a} Pw1Fn t) u S n) ↔ t(u = {t} ({a} Pw1Fn t u S n)))
73, 4, 63bitr2i 264 . . . . 5 (({{a}} SI Pw1Fn u u S n) ↔ t(u = {t} ({a} Pw1Fn t u S n)))
87exbii 1582 . . . 4 (u({{a}} SI Pw1Fn u u S n) ↔ ut(u = {t} ({a} Pw1Fn t u S n)))
9 excom 1741 . . . 4 (ut(u = {t} ({a} Pw1Fn t u S n)) ↔ tu(u = {t} ({a} Pw1Fn t u S n)))
108, 9bitri 240 . . 3 (u({{a}} SI Pw1Fn u u S n) ↔ tu(u = {t} ({a} Pw1Fn t u S n)))
11 snex 4111 . . . . . 6 {t} V
12 breq1 4642 . . . . . . 7 (u = {t} → (u S n ↔ {t} S n))
1312anbi2d 684 . . . . . 6 (u = {t} → (({a} Pw1Fn t u S n) ↔ ({a} Pw1Fn t {t} S n)))
1411, 13ceqsexv 2894 . . . . 5 (u(u = {t} ({a} Pw1Fn t u S n)) ↔ ({a} Pw1Fn t {t} S n))
15 vex 2862 . . . . . . 7 a V
1615brpw1fn 5854 . . . . . 6 ({a} Pw1Fn tt = 1a)
17 vex 2862 . . . . . . 7 t V
18 vex 2862 . . . . . . 7 n V
1917, 18brssetsn 4759 . . . . . 6 ({t} S nt n)
2016, 19anbi12i 678 . . . . 5 (({a} Pw1Fn t {t} S n) ↔ (t = 1a t n))
2114, 20bitri 240 . . . 4 (u(u = {t} ({a} Pw1Fn t u S n)) ↔ (t = 1a t n))
2221exbii 1582 . . 3 (tu(u = {t} ({a} Pw1Fn t u S n)) ↔ t(t = 1a t n))
2310, 22bitri 240 . 2 (u({{a}} SI Pw1Fn u u S n) ↔ t(t = 1a t n))
24 opelco 4884 . 2 ({{a}}, n ( S SI Pw1Fn ) ↔ u({{a}} SI Pw1Fn u u S n))
25 df-clel 2349 . 2 (1a nt(t = 1a t n))
2623, 24, 253bitr4i 268 1 ({{a}}, n ( S SI Pw1Fn ) ↔ 1a n)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {csn 3737  ℘1cpw1 4135  ⟨cop 4561   class class class wbr 4639   S csset 4719   SI csi 4720   ∘ ccom 4721   Pw1Fn cpw1fn 5765 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-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-fv 4795  df-mpt 5652  df-pw1fn 5766 This theorem is referenced by:  ceex  6174
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