Theorem r1om 10236

Description: The set of hereditarily finite sets is countable. See ackbij2 10235 for an explicit bijection that works without Infinity. See also r1omALT 10768. (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 9635	. . . 4 ⊢ ω ∈ V
2		limom 7868	. . . 4 ⊢ Lim ω
3		r1lim 9764	. . . 4 ⊢ ((ω ∈ V ∧ Lim ω) → (𝑅1‘ω) = ∪ 𝑎 ∈ ω (𝑅1‘𝑎))
4	1, 2, 3	mp2an 691	. . 3 ⊢ (𝑅1‘ω) = ∪ 𝑎 ∈ ω (𝑅1‘𝑎)
5		r1fnon 9759	. . . 4 ⊢ 𝑅1 Fn On
6		fnfun 6647	. . . 4 ⊢ (𝑅1 Fn On → Fun 𝑅1)
7		funiunfv 7244	. . . 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 2761	. 2 ⊢ (𝑅1‘ω) = ∪ (𝑅1 “ ω)
10		iuneq1 5013	. . . . . . 7 ⊢ (𝑒 = 𝑎 → ∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓) = ∪ 𝑓 ∈ 𝑎 ({𝑓} × 𝒫 𝑓))
11		sneq 4638	. . . . . . . . 9 ⊢ (𝑓 = 𝑏 → {𝑓} = {𝑏})
12		pweq 4616	. . . . . . . . 9 ⊢ (𝑓 = 𝑏 → 𝒫 𝑓 = 𝒫 𝑏)
13	11, 12	xpeq12d 5707	. . . . . . . 8 ⊢ (𝑓 = 𝑏 → ({𝑓} × 𝒫 𝑓) = ({𝑏} × 𝒫 𝑏))
14	13	cbviunv 5043	. . . . . . 7 ⊢ ∪ 𝑓 ∈ 𝑎 ({𝑓} × 𝒫 𝑓) = ∪ 𝑏 ∈ 𝑎 ({𝑏} × 𝒫 𝑏)
15	10, 14	eqtrdi 2789	. . . . . 6 ⊢ (𝑒 = 𝑎 → ∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓) = ∪ 𝑏 ∈ 𝑎 ({𝑏} × 𝒫 𝑏))
16	15	fveq2d 6893	. . . . 5 ⊢ (𝑒 = 𝑎 → (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)) = (card‘∪ 𝑏 ∈ 𝑎 ({𝑏} × 𝒫 𝑏)))
17	16	cbvmptv 5261	. . . 4 ⊢ (𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓))) = (𝑎 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑏 ∈ 𝑎 ({𝑏} × 𝒫 𝑏)))
18		dmeq 5902	. . . . . . . 8 ⊢ (𝑐 = 𝑎 → dom 𝑐 = dom 𝑎)
19	18	pweqd 4619	. . . . . . 7 ⊢ (𝑐 = 𝑎 → 𝒫 dom 𝑐 = 𝒫 dom 𝑎)
20		imaeq1 6053	. . . . . . . 8 ⊢ (𝑐 = 𝑎 → (𝑐 “ 𝑑) = (𝑎 “ 𝑑))
21	20	fveq2d 6893	. . . . . . 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 5239	. . . . . 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 6054	. . . . . . . 8 ⊢ (𝑑 = 𝑏 → (𝑎 “ 𝑑) = (𝑎 “ 𝑏))
24	23	fveq2d 6893	. . . . . . 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 5261	. . . . . 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 2789	. . . . 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 5261	. . . 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 2733	. . . 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 10235	. . 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 6902	. . . . 5 ⊢ (𝑅1‘ω) ∈ V
31	9, 30	eqeltrri 2831	. . . 4 ⊢ ∪ (𝑅1 “ ω) ∈ V
32	31	f1oen 8966	. . 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 5169	1 ⊢ (𝑅1‘ω) ≈ ω

Syntax hints:   = wceq 1542   ∈ wcel 2107  Vcvv 3475   ∩ cin 3947  ∅c0 4322  𝒫 cpw 4602  {csn 4628  ∪ cuni 4908  ∪ ciun 4997   class class class wbr 5148   ↦ cmpt 5231   × cxp 5674  dom cdm 5676   “ cima 5679  Oncon0 6362  Lim wlim 6363  Fun wfun 6535   Fn wfn 6536  –1-1-onto→wf1o 6540  ‘cfv 6541  ωcom 7852  reccrdg 8406   ≈ cen 8933  Fincfn 8936  𝑅₁cr1 9754  cardccrd 9927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-inf2 9633

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-2o 8464  df-oadd 8467  df-er 8700  df-map 8819  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-r1 9756  df-rank 9757  df-dju 9893  df-card 9931

This theorem is referenced by: (None)