Theorem elimapw13 4947

Description: Membership in an image under three unit power classes. (Contributed by set.mm contributors, 18-Mar-2015.)

Assertion

Ref	Expression
elimapw13	|- (A e. (B"~P1 ~P1 ~P1 C) <-> E.x e. C <.{{{x}}}, A>. e. B)

Distinct variable groups:   x,A   x,B   x,C

Proof of Theorem elimapw13

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elimapw12 4946	. 2 |- (A e. (B"~P1 ~P1 ~P1 C) <-> E.t e. ~P1 C<.{{t}}, A>. e. B)
2		df-rex 2621	. . 3 |- (E.t e. ~P1 C<.{{t}}, A>. e. B <-> E.t(t e. ~P1 C /\ <.{{t}}, A>. e. B))
3		elpw1 4145	. . . . . . . 8 |- (t e. ~P1 C <-> E.x e. C t = {x})
4	3	anbi1i 676	. . . . . . 7 |- ((t e. ~P1 C /\ <.{{t}}, A>. e. B) <-> (E.x e. C t = {x} /\ <.{{t}}, A>. e. B))
5		r19.41v 2765	. . . . . . 7 |- (E.x e. C (t = {x} /\ <.{{t}}, A>. e. B) <-> (E.x e. C t = {x} /\ <.{{t}}, A>. e. B))
6	4, 5	bitr4i 243	. . . . . 6 |- ((t e. ~P1 C /\ <.{{t}}, A>. e. B) <-> E.x e. C (t = {x} /\ <.{{t}}, A>. e. B))
7	6	exbii 1582	. . . . 5 |- (E.t(t e. ~P1 C /\ <.{{t}}, A>. e. B) <-> E.tE.x e. C (t = {x} /\ <.{{t}}, A>. e. B))
8		rexcom4 2879	. . . . 5 |- (E.x e. C E.t(t = {x} /\ <.{{t}}, A>. e. B) <-> E.tE.x e. C (t = {x} /\ <.{{t}}, A>. e. B))
9	7, 8	bitr4i 243	. . . 4 |- (E.t(t e. ~P1 C /\ <.{{t}}, A>. e. B) <-> E.x e. C E.t(t = {x} /\ <.{{t}}, A>. e. B))
10		snex 4112	. . . . . 6 |- {x} e. _V
11		sneq 3745	. . . . . . . . 9 |- (t = {x} -> {t} = {{x}})
12	11	sneqd 3747	. . . . . . . 8 |- (t = {x} -> {{t}} = {{{x}}})
13	12	opeq1d 4585	. . . . . . 7 |- (t = {x} -> <.{{t}}, A>. = <.{{{x}}}, A>.)
14	13	eleq1d 2419	. . . . . 6 |- (t = {x} -> (<.{{t}}, A>. e. B <-> <.{{{x}}}, A>. e. B))
15	10, 14	ceqsexv 2895	. . . . 5 |- (E.t(t = {x} /\ <.{{t}}, A>. e. B) <-> <.{{{x}}}, A>. e. B)
16	15	rexbii 2640	. . . 4 |- (E.x e. C E.t(t = {x} /\ <.{{t}}, A>. e. B) <-> E.x e. C <.{{{x}}}, A>. e. B)
17	9, 16	bitri 240	. . 3 |- (E.t(t e. ~P1 C /\ <.{{t}}, A>. e. B) <-> E.x e. C <.{{{x}}}, A>. e. B)
18	2, 17	bitri 240	. 2 |- (E.t e. ~P1 C<.{{t}}, A>. e. B <-> E.x e. C <.{{{x}}}, A>. e. B)
19	1, 18	bitri 240	1 |- (A e. (B"~P1 ~P1 ~P1 C) <-> E.x e. C <.{{{x}}}, A>. e. B)

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  E.wrex 2616  {csn 3738  ~P₁ cpw1 4136  <.cop 4562  "cima 4723

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-br 4641  df-ima 4728

This theorem is referenced by:  nchoicelem11  6300  nchoicelem16  6305