Theorem r1om 10196

Description: The set of hereditarily finite sets is countable. See ackbij2 10195 for an explicit bijection that works without Infinity. See also r1omALT 10729. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Assertion

Ref	Expression
r1om	⊢ (𝑅1‘ω) ≈ ω

Proof of Theorem r1om

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		omex 9596	. . . 4 ⊢ ω ∈ V
2		limom 7858	. . . 4 ⊢ Lim ω
3		r1lim 9725	. . . 4 ⊢ ((ω ∈ V ∧ Lim ω) → (𝑅1‘ω) = ∪ 𝑎 ∈ ω (𝑅1‘𝑎))
4	1, 2, 3	mp2an 692	. . 3 ⊢ (𝑅1‘ω) = ∪ 𝑎 ∈ ω (𝑅1‘𝑎)
5		r1fnon 9720	. . . 4 ⊢ 𝑅1 Fn On
6		fnfun 6618	. . . 4 ⊢ (𝑅1 Fn On → Fun 𝑅1)
7		funiunfv 7222	. . . 4 ⊢ (Fun 𝑅1 → ∪ 𝑎 ∈ ω (𝑅1‘𝑎) = ∪ (𝑅1 “ ω))
8	5, 6, 7	mp2b 10	. . 3 ⊢ ∪ 𝑎 ∈ ω (𝑅1‘𝑎) = ∪ (𝑅1 “ ω)
9	4, 8	eqtri 2752	. 2 ⊢ (𝑅1‘ω) = ∪ (𝑅1 “ ω)
10		iuneq1 4972	. . . . . . 7 ⊢ (𝑒 = 𝑎 → ∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓) = ∪ 𝑓 ∈ 𝑎 ({𝑓} × 𝒫 𝑓))
11		sneq 4599	. . . . . . . . 9 ⊢ (𝑓 = 𝑏 → {𝑓} = {𝑏})
12		pweq 4577	. . . . . . . . 9 ⊢ (𝑓 = 𝑏 → 𝒫 𝑓 = 𝒫 𝑏)
13	11, 12	xpeq12d 5669	. . . . . . . 8 ⊢ (𝑓 = 𝑏 → ({𝑓} × 𝒫 𝑓) = ({𝑏} × 𝒫 𝑏))
14	13	cbviunv 5004	. . . . . . 7 ⊢ ∪ 𝑓 ∈ 𝑎 ({𝑓} × 𝒫 𝑓) = ∪ 𝑏 ∈ 𝑎 ({𝑏} × 𝒫 𝑏)
15	10, 14	eqtrdi 2780	. . . . . 6 ⊢ (𝑒 = 𝑎 → ∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓) = ∪ 𝑏 ∈ 𝑎 ({𝑏} × 𝒫 𝑏))
16	15	fveq2d 6862	. . . . 5 ⊢ (𝑒 = 𝑎 → (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)) = (card‘∪ 𝑏 ∈ 𝑎 ({𝑏} × 𝒫 𝑏)))
17	16	cbvmptv 5211	. . . 4 ⊢ (𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓))) = (𝑎 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑏 ∈ 𝑎 ({𝑏} × 𝒫 𝑏)))
18		dmeq 5867	. . . . . . . 8 ⊢ (𝑐 = 𝑎 → dom 𝑐 = dom 𝑎)
19	18	pweqd 4580	. . . . . . 7 ⊢ (𝑐 = 𝑎 → 𝒫 dom 𝑐 = 𝒫 dom 𝑎)
20		imaeq1 6026	. . . . . . . 8 ⊢ (𝑐 = 𝑎 → (𝑐 “ 𝑑) = (𝑎 “ 𝑑))
21	20	fveq2d 6862	. . . . . . 7 ⊢ (𝑐 = 𝑎 → ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑐 “ 𝑑)) = ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎 “ 𝑑)))
22	19, 21	mpteq12dv 5194	. . . . . 6 ⊢ (𝑐 = 𝑎 → (𝑑 ∈ 𝒫 dom 𝑐 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑐 “ 𝑑))) = (𝑑 ∈ 𝒫 dom 𝑎 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎 “ 𝑑))))
23		imaeq2 6027	. . . . . . . 8 ⊢ (𝑑 = 𝑏 → (𝑎 “ 𝑑) = (𝑎 “ 𝑏))
24	23	fveq2d 6862	. . . . . . 7 ⊢ (𝑑 = 𝑏 → ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎 “ 𝑑)) = ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎 “ 𝑏)))
25	24	cbvmptv 5211	. . . . . 6 ⊢ (𝑑 ∈ 𝒫 dom 𝑎 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎 “ 𝑑))) = (𝑏 ∈ 𝒫 dom 𝑎 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎 “ 𝑏)))
26	22, 25	eqtrdi 2780	. . . . 5 ⊢ (𝑐 = 𝑎 → (𝑑 ∈ 𝒫 dom 𝑐 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑐 “ 𝑑))) = (𝑏 ∈ 𝒫 dom 𝑎 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎 “ 𝑏))))
27	26	cbvmptv 5211	. . . 4 ⊢ (𝑐 ∈ V ↦ (𝑑 ∈ 𝒫 dom 𝑐 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑐 “ 𝑑)))) = (𝑎 ∈ V ↦ (𝑏 ∈ 𝒫 dom 𝑎 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑎 “ 𝑏))))
28		eqid 2729	. . . 4 ⊢ ∪ (rec((𝑐 ∈ V ↦ (𝑑 ∈ 𝒫 dom 𝑐 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑐 “ 𝑑)))), ∅) “ ω) = ∪ (rec((𝑐 ∈ V ↦ (𝑑 ∈ 𝒫 dom 𝑐 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑐 “ 𝑑)))), ∅) “ ω)
29	17, 27, 28	ackbij2 10195	. . 3 ⊢ ∪ (rec((𝑐 ∈ V ↦ (𝑑 ∈ 𝒫 dom 𝑐 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑐 “ 𝑑)))), ∅) “ ω):∪ (𝑅1 “ ω)–1-1-onto→ω
30		fvex 6871	. . . . 5 ⊢ (𝑅1‘ω) ∈ V
31	9, 30	eqeltrri 2825	. . . 4 ⊢ ∪ (𝑅1 “ ω) ∈ V
32	31	f1oen 8944	. . 3 ⊢ (∪ (rec((𝑐 ∈ V ↦ (𝑑 ∈ 𝒫 dom 𝑐 ↦ ((𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)))‘(𝑐 “ 𝑑)))), ∅) “ ω):∪ (𝑅1 “ ω)–1-1-onto→ω → ∪ (𝑅1 “ ω) ≈ ω)
33	29, 32	ax-mp 5	. 2 ⊢ ∪ (𝑅1 “ ω) ≈ ω
34	9, 33	eqbrtri 5128	1 ⊢ (𝑅1‘ω) ≈ ω

Syntax hints:   = wceq 1540   ∈ wcel 2109  Vcvv 3447   ∩ cin 3913  ∅c0 4296  𝒫 cpw 4563  {csn 4589  ∪ cuni 4871  ∪ ciun 4955   class class class wbr 5107   ↦ cmpt 5188   × cxp 5636  dom cdm 5638   “ cima 5641  Oncon0 6332  Lim wlim 6333  Fun wfun 6505   Fn wfn 6506  –1-1-onto→wf1o 6510  ‘cfv 6511  ωcom 7842  reccrdg 8377   ≈ cen 8915  Fincfn 8918  𝑅₁cr1 9715  cardccrd 9888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-inf2 9594

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-oadd 8438  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-r1 9717  df-rank 9718  df-dju 9854  df-card 9892

This theorem is referenced by: (None)