Theorem pw1fnex 5853

Description: The unit power class function is a set. (Contributed by SF, 25-Feb-2015.)

Assertion

Ref	Expression
pw1fnex	⊢ Pw1Fn ∈ V

Proof of Theorem pw1fnex

Dummy variables x y t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-pw1fn 5767	. . 3 ⊢ Pw1Fn = (x ∈ 1c ↦ ℘1∪x)
2		oteltxp 5783	. . . . . . . 8 ⊢ (⟨{{t}}, ⟨{y}, x⟩⟩ ∈ ( I ⊗ (( SI ◡ S ⊗ S ) “ 1c)) ↔ (⟨{{t}}, {y}⟩ ∈ I ∧ ⟨{{t}}, x⟩ ∈ (( SI ◡ S ⊗ S ) “ 1c)))
3		snex 4112	. . . . . . . . . . 11 ⊢ {y} ∈ V
4	3	ideq 4871	. . . . . . . . . 10 ⊢ ({{t}} I {y} ↔ {{t}} = {y})
5		df-br 4641	. . . . . . . . . 10 ⊢ ({{t}} I {y} ↔ ⟨{{t}}, {y}⟩ ∈ I )
6		eqcom 2355	. . . . . . . . . . 11 ⊢ ({{t}} = {y} ↔ {y} = {{t}})
7		vex 2863	. . . . . . . . . . . 12 ⊢ y ∈ V
8	7	sneqb 3877	. . . . . . . . . . 11 ⊢ ({y} = {{t}} ↔ y = {t})
9	6, 8	bitri 240	. . . . . . . . . 10 ⊢ ({{t}} = {y} ↔ y = {t})
10	4, 5, 9	3bitr3i 266	. . . . . . . . 9 ⊢ (⟨{{t}}, {y}⟩ ∈ I ↔ y = {t})
11		oteltxp 5783	. . . . . . . . . . . 12 ⊢ (⟨{y}, ⟨{{t}}, x⟩⟩ ∈ ( SI ◡ S ⊗ S ) ↔ (⟨{y}, {{t}}⟩ ∈ SI ◡ S ∧ ⟨{y}, x⟩ ∈ S ))
12		snex 4112	. . . . . . . . . . . . . . 15 ⊢ {t} ∈ V
13	7, 12	brsnsi 5774	. . . . . . . . . . . . . 14 ⊢ ({y} SI ◡ S {{t}} ↔ y◡ S {t})
14		df-br 4641	. . . . . . . . . . . . . 14 ⊢ ({y} SI ◡ S {{t}} ↔ ⟨{y}, {{t}}⟩ ∈ SI ◡ S )
15		brcnv 4893	. . . . . . . . . . . . . . 15 ⊢ (y◡ S {t} ↔ {t} S y)
16		vex 2863	. . . . . . . . . . . . . . . 16 ⊢ t ∈ V
17	16, 7	brssetsn 4760	. . . . . . . . . . . . . . 15 ⊢ ({t} S y ↔ t ∈ y)
18	15, 17	bitri 240	. . . . . . . . . . . . . 14 ⊢ (y◡ S {t} ↔ t ∈ y)
19	13, 14, 18	3bitr3i 266	. . . . . . . . . . . . 13 ⊢ (⟨{y}, {{t}}⟩ ∈ SI ◡ S ↔ t ∈ y)
20		vex 2863	. . . . . . . . . . . . . 14 ⊢ x ∈ V
21	7, 20	opelssetsn 4761	. . . . . . . . . . . . 13 ⊢ (⟨{y}, x⟩ ∈ S ↔ y ∈ x)
22	19, 21	anbi12i 678	. . . . . . . . . . . 12 ⊢ ((⟨{y}, {{t}}⟩ ∈ SI ◡ S ∧ ⟨{y}, x⟩ ∈ S ) ↔ (t ∈ y ∧ y ∈ x))
23	11, 22	bitri 240	. . . . . . . . . . 11 ⊢ (⟨{y}, ⟨{{t}}, x⟩⟩ ∈ ( SI ◡ S ⊗ S ) ↔ (t ∈ y ∧ y ∈ x))
24	23	exbii 1582	. . . . . . . . . 10 ⊢ (∃y⟨{y}, ⟨{{t}}, x⟩⟩ ∈ ( SI ◡ S ⊗ S ) ↔ ∃y(t ∈ y ∧ y ∈ x))
25		elima1c 4948	. . . . . . . . . 10 ⊢ (⟨{{t}}, x⟩ ∈ (( SI ◡ S ⊗ S ) “ 1c) ↔ ∃y⟨{y}, ⟨{{t}}, x⟩⟩ ∈ ( SI ◡ S ⊗ S ))
26		eluni 3895	. . . . . . . . . 10 ⊢ (t ∈ ∪x ↔ ∃y(t ∈ y ∧ y ∈ x))
27	24, 25, 26	3bitr4i 268	. . . . . . . . 9 ⊢ (⟨{{t}}, x⟩ ∈ (( SI ◡ S ⊗ S ) “ 1c) ↔ t ∈ ∪x)
28	10, 27	anbi12i 678	. . . . . . . 8 ⊢ ((⟨{{t}}, {y}⟩ ∈ I ∧ ⟨{{t}}, x⟩ ∈ (( SI ◡ S ⊗ S ) “ 1c)) ↔ (y = {t} ∧ t ∈ ∪x))
29	2, 28	bitri 240	. . . . . . 7 ⊢ (⟨{{t}}, ⟨{y}, x⟩⟩ ∈ ( I ⊗ (( SI ◡ S ⊗ S ) “ 1c)) ↔ (y = {t} ∧ t ∈ ∪x))
30		ancom 437	. . . . . . 7 ⊢ ((y = {t} ∧ t ∈ ∪x) ↔ (t ∈ ∪x ∧ y = {t}))
31	29, 30	bitri 240	. . . . . 6 ⊢ (⟨{{t}}, ⟨{y}, x⟩⟩ ∈ ( I ⊗ (( SI ◡ S ⊗ S ) “ 1c)) ↔ (t ∈ ∪x ∧ y = {t}))
32	31	exbii 1582	. . . . 5 ⊢ (∃t⟨{{t}}, ⟨{y}, x⟩⟩ ∈ ( I ⊗ (( SI ◡ S ⊗ S ) “ 1c)) ↔ ∃t(t ∈ ∪x ∧ y = {t}))
33		elimapw11c 4949	. . . . 5 ⊢ (⟨{y}, x⟩ ∈ (( I ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c) ↔ ∃t⟨{{t}}, ⟨{y}, x⟩⟩ ∈ ( I ⊗ (( SI ◡ S ⊗ S ) “ 1c)))
34		elpw1 4145	. . . . . 6 ⊢ (y ∈ ℘1∪x ↔ ∃t ∈ ∪xy = {t})
35		df-rex 2621	. . . . . 6 ⊢ (∃t ∈ ∪xy = {t} ↔ ∃t(t ∈ ∪x ∧ y = {t}))
36	34, 35	bitri 240	. . . . 5 ⊢ (y ∈ ℘1∪x ↔ ∃t(t ∈ ∪x ∧ y = {t}))
37	32, 33, 36	3bitr4i 268	. . . 4 ⊢ (⟨{y}, x⟩ ∈ (( I ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c) ↔ y ∈ ℘1∪x)
38	37	releqmpt 5809	. . 3 ⊢ ((1c × V) ∩ ◡ ∼ (( Ins3 S ⊕ Ins2 (( I ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) “ 1c)) = (x ∈ 1c ↦ ℘1∪x)
39	1, 38	eqtr4i 2376	. 2 ⊢ Pw1Fn = ((1c × V) ∩ ◡ ∼ (( Ins3 S ⊕ Ins2 (( I ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) “ 1c))
40		1cex 4143	. . 3 ⊢ 1c ∈ V
41		idex 5505	. . . . 5 ⊢ I ∈ V
42		ssetex 4745	. . . . . . . . 9 ⊢ S ∈ V
43	42	cnvex 5103	. . . . . . . 8 ⊢ ◡ S ∈ V
44	43	siex 4754	. . . . . . 7 ⊢ SI ◡ S ∈ V
45	44, 42	txpex 5786	. . . . . 6 ⊢ ( SI ◡ S ⊗ S ) ∈ V
46	45, 40	imaex 4748	. . . . 5 ⊢ (( SI ◡ S ⊗ S ) “ 1c) ∈ V
47	41, 46	txpex 5786	. . . 4 ⊢ ( I ⊗ (( SI ◡ S ⊗ S ) “ 1c)) ∈ V
48	40	pw1ex 4304	. . . 4 ⊢ ℘11c ∈ V
49	47, 48	imaex 4748	. . 3 ⊢ (( I ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c) ∈ V
50	40, 49	mptexlem 5811	. 2 ⊢ ((1c × V) ∩ ◡ ∼ (( Ins3 S ⊕ Ins2 (( I ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) “ 1c)) ∈ V
51	39, 50	eqeltri 2423	1 ⊢ Pw1Fn ∈ V

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  Vcvv 2860   ∼ ccompl 3206   ∩ cin 3209   ⊕ csymdif 3210  {csn 3738  ∪cuni 3892  1_cc1c 4135  ℘₁cpw1 4136  ⟨cop 4562   class class class wbr 4640   S csset 4720   SI csi 4721   “ cima 4723   I cid 4764   × cxp 4771  ^◡ccnv 4772   ↦ cmpt 5652   ⊗ ctxp 5736   Ins2 cins2 5750   Ins3 cins3 5752   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-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-2nd 4798  df-mpt 5653  df-txp 5737  df-ins2 5751  df-ins3 5753  df-pw1fn 5767

This theorem is referenced by:  enpw1pw  6076  ovcelem1  6172  ceex  6175  tcfnex  6245