Theorem pw1fnf1o 5856

Description: Pw1Fn is a one-to-one function with domain 1_c and range ℘1_c. (Contributed by SF, 26-Feb-2015.)

Assertion

Ref	Expression
pw1fnf1o	⊢ Pw1Fn :1c–1-1-onto→℘1c

Proof of Theorem pw1fnf1o

Dummy variables a b x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnpw1fn 5854	. 2 ⊢ Pw1Fn Fn 1c
2		df-pw1fn 5767	. . . 4 ⊢ Pw1Fn = (x ∈ 1c ↦ ℘1∪x)
3	2	rnmpt 5687	. . 3 ⊢ ran Pw1Fn = {y ∣ ∃x ∈ 1c y = ℘1∪x}
4		vex 2863	. . . . . . 7 ⊢ y ∈ V
5	4	sspw1 4336	. . . . . 6 ⊢ (y ⊆ ℘1V ↔ ∃z(z ⊆ V ∧ y = ℘1z))
6		df1c2 4169	. . . . . . 7 ⊢ 1c = ℘1V
7	6	sseq2i 3297	. . . . . 6 ⊢ (y ⊆ 1c ↔ y ⊆ ℘1V)
8		ssv 3292	. . . . . . . 8 ⊢ z ⊆ V
9	8	biantrur 492	. . . . . . 7 ⊢ (y = ℘1z ↔ (z ⊆ V ∧ y = ℘1z))
10	9	exbii 1582	. . . . . 6 ⊢ (∃z y = ℘1z ↔ ∃z(z ⊆ V ∧ y = ℘1z))
11	5, 7, 10	3bitr4i 268	. . . . 5 ⊢ (y ⊆ 1c ↔ ∃z y = ℘1z)
12	4	elpw 3729	. . . . 5 ⊢ (y ∈ ℘1c ↔ y ⊆ 1c)
13		df-rex 2621	. . . . . 6 ⊢ (∃x ∈ 1c y = ℘1∪x ↔ ∃x(x ∈ 1c ∧ y = ℘1∪x))
14		el1c 4140	. . . . . . . . 9 ⊢ (x ∈ 1c ↔ ∃z x = {z})
15	14	anbi1i 676	. . . . . . . 8 ⊢ ((x ∈ 1c ∧ y = ℘1∪x) ↔ (∃z x = {z} ∧ y = ℘1∪x))
16		19.41v 1901	. . . . . . . 8 ⊢ (∃z(x = {z} ∧ y = ℘1∪x) ↔ (∃z x = {z} ∧ y = ℘1∪x))
17	15, 16	bitr4i 243	. . . . . . 7 ⊢ ((x ∈ 1c ∧ y = ℘1∪x) ↔ ∃z(x = {z} ∧ y = ℘1∪x))
18	17	exbii 1582	. . . . . 6 ⊢ (∃x(x ∈ 1c ∧ y = ℘1∪x) ↔ ∃x∃z(x = {z} ∧ y = ℘1∪x))
19		excom 1741	. . . . . . 7 ⊢ (∃x∃z(x = {z} ∧ y = ℘1∪x) ↔ ∃z∃x(x = {z} ∧ y = ℘1∪x))
20		snex 4112	. . . . . . . . 9 ⊢ {z} ∈ V
21		unieq 3901	. . . . . . . . . . . 12 ⊢ (x = {z} → ∪x = ∪{z})
22		vex 2863	. . . . . . . . . . . . 13 ⊢ z ∈ V
23	22	unisn 3908	. . . . . . . . . . . 12 ⊢ ∪{z} = z
24	21, 23	syl6eq 2401	. . . . . . . . . . 11 ⊢ (x = {z} → ∪x = z)
25		pw1eq 4144	. . . . . . . . . . 11 ⊢ (∪x = z → ℘1∪x = ℘1z)
26	24, 25	syl 15	. . . . . . . . . 10 ⊢ (x = {z} → ℘1∪x = ℘1z)
27	26	eqeq2d 2364	. . . . . . . . 9 ⊢ (x = {z} → (y = ℘1∪x ↔ y = ℘1z))
28	20, 27	ceqsexv 2895	. . . . . . . 8 ⊢ (∃x(x = {z} ∧ y = ℘1∪x) ↔ y = ℘1z)
29	28	exbii 1582	. . . . . . 7 ⊢ (∃z∃x(x = {z} ∧ y = ℘1∪x) ↔ ∃z y = ℘1z)
30	19, 29	bitri 240	. . . . . 6 ⊢ (∃x∃z(x = {z} ∧ y = ℘1∪x) ↔ ∃z y = ℘1z)
31	13, 18, 30	3bitri 262	. . . . 5 ⊢ (∃x ∈ 1c y = ℘1∪x ↔ ∃z y = ℘1z)
32	11, 12, 31	3bitr4i 268	. . . 4 ⊢ (y ∈ ℘1c ↔ ∃x ∈ 1c y = ℘1∪x)
33	32	abbi2i 2465	. . 3 ⊢ ℘1c = {y ∣ ∃x ∈ 1c y = ℘1∪x}
34	3, 33	eqtr4i 2376	. 2 ⊢ ran Pw1Fn = ℘1c
35		el1c 4140	. . . . . 6 ⊢ (x ∈ 1c ↔ ∃a x = {a})
36		el1c 4140	. . . . . 6 ⊢ (y ∈ 1c ↔ ∃b y = {b})
37	35, 36	anbi12i 678	. . . . 5 ⊢ ((x ∈ 1c ∧ y ∈ 1c) ↔ (∃a x = {a} ∧ ∃b y = {b}))
38		eeanv 1913	. . . . 5 ⊢ (∃a∃b(x = {a} ∧ y = {b}) ↔ (∃a x = {a} ∧ ∃b y = {b}))
39	37, 38	bitr4i 243	. . . 4 ⊢ ((x ∈ 1c ∧ y ∈ 1c) ↔ ∃a∃b(x = {a} ∧ y = {b}))
40		pw111 4171	. . . . . . . 8 ⊢ (℘1a = ℘1b ↔ a = b)
41	40	biimpi 186	. . . . . . 7 ⊢ (℘1a = ℘1b → a = b)
42	41	a1i 10	. . . . . 6 ⊢ ((x = {a} ∧ y = {b}) → (℘1a = ℘1b → a = b))
43		fveq2 5329	. . . . . . . 8 ⊢ (x = {a} → ( Pw1Fn ‘x) = ( Pw1Fn ‘{a}))
44		vex 2863	. . . . . . . . 9 ⊢ a ∈ V
45	44	pw1fnval 5852	. . . . . . . 8 ⊢ ( Pw1Fn ‘{a}) = ℘1a
46	43, 45	syl6eq 2401	. . . . . . 7 ⊢ (x = {a} → ( Pw1Fn ‘x) = ℘1a)
47		fveq2 5329	. . . . . . . 8 ⊢ (y = {b} → ( Pw1Fn ‘y) = ( Pw1Fn ‘{b}))
48		vex 2863	. . . . . . . . 9 ⊢ b ∈ V
49	48	pw1fnval 5852	. . . . . . . 8 ⊢ ( Pw1Fn ‘{b}) = ℘1b
50	47, 49	syl6eq 2401	. . . . . . 7 ⊢ (y = {b} → ( Pw1Fn ‘y) = ℘1b)
51	46, 50	eqeqan12d 2368	. . . . . 6 ⊢ ((x = {a} ∧ y = {b}) → (( Pw1Fn ‘x) = ( Pw1Fn ‘y) ↔ ℘1a = ℘1b))
52		eqeq12 2365	. . . . . . 7 ⊢ ((x = {a} ∧ y = {b}) → (x = y ↔ {a} = {b}))
53	44	sneqb 3877	. . . . . . 7 ⊢ ({a} = {b} ↔ a = b)
54	52, 53	syl6bb 252	. . . . . 6 ⊢ ((x = {a} ∧ y = {b}) → (x = y ↔ a = b))
55	42, 51, 54	3imtr4d 259	. . . . 5 ⊢ ((x = {a} ∧ y = {b}) → (( Pw1Fn ‘x) = ( Pw1Fn ‘y) → x = y))
56	55	exlimivv 1635	. . . 4 ⊢ (∃a∃b(x = {a} ∧ y = {b}) → (( Pw1Fn ‘x) = ( Pw1Fn ‘y) → x = y))
57	39, 56	sylbi 187	. . 3 ⊢ ((x ∈ 1c ∧ y ∈ 1c) → (( Pw1Fn ‘x) = ( Pw1Fn ‘y) → x = y))
58	57	rgen2a 2681	. 2 ⊢ ∀x ∈ 1c ∀y ∈ 1c (( Pw1Fn ‘x) = ( Pw1Fn ‘y) → x = y)
59		dff1o6 5476	. 2 ⊢ ( Pw1Fn :1c–1-1-onto→℘1c ↔ ( Pw1Fn Fn 1c ∧ ran Pw1Fn = ℘1c ∧ ∀x ∈ 1c ∀y ∈ 1c (( Pw1Fn ‘x) = ( Pw1Fn ‘y) → x = y)))
60	1, 34, 58, 59	mpbir3an 1134	1 ⊢ Pw1Fn :1c–1-1-onto→℘1c

Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2615  ∃wrex 2616  Vcvv 2860   ⊆ wss 3258  ℘cpw 3723  {csn 3738  ∪cuni 3892  1_cc1c 4135  ℘₁cpw1 4136  ran crn 4774   Fn wfn 4777  –1-1-onto→wf1o 4781   ‘cfv 4782   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-co 4727  df-ima 4728  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-mpt 5653  df-pw1fn 5767

This theorem is referenced by:  enpw1pw  6076