Theorem r1om 10257

Description: The set of hereditarily finite sets is countable. See ackbij2 10256 for an explicit bijection that works without Infinity. See also r1omALT 10790. (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 9657	. . . 4 ⊢ ω ∈ V
2		limom 7877	. . . 4 ⊢ Lim ω
3		r1lim 9786	. . . 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 9781	. . . 4 ⊢ 𝑅1 Fn On
6		fnfun 6638	. . . 4 ⊢ (𝑅1 Fn On → Fun 𝑅1)
7		funiunfv 7240	. . . 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 2758	. 2 ⊢ (𝑅1‘ω) = ∪ (𝑅1 “ ω)
10		iuneq1 4984	. . . . . . 7 ⊢ (𝑒 = 𝑎 → ∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓) = ∪ 𝑓 ∈ 𝑎 ({𝑓} × 𝒫 𝑓))
11		sneq 4611	. . . . . . . . 9 ⊢ (𝑓 = 𝑏 → {𝑓} = {𝑏})
12		pweq 4589	. . . . . . . . 9 ⊢ (𝑓 = 𝑏 → 𝒫 𝑓 = 𝒫 𝑏)
13	11, 12	xpeq12d 5685	. . . . . . . 8 ⊢ (𝑓 = 𝑏 → ({𝑓} × 𝒫 𝑓) = ({𝑏} × 𝒫 𝑏))
14	13	cbviunv 5016	. . . . . . 7 ⊢ ∪ 𝑓 ∈ 𝑎 ({𝑓} × 𝒫 𝑓) = ∪ 𝑏 ∈ 𝑎 ({𝑏} × 𝒫 𝑏)
15	10, 14	eqtrdi 2786	. . . . . 6 ⊢ (𝑒 = 𝑎 → ∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓) = ∪ 𝑏 ∈ 𝑎 ({𝑏} × 𝒫 𝑏))
16	15	fveq2d 6880	. . . . 5 ⊢ (𝑒 = 𝑎 → (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)) = (card‘∪ 𝑏 ∈ 𝑎 ({𝑏} × 𝒫 𝑏)))
17	16	cbvmptv 5225	. . . 4 ⊢ (𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓))) = (𝑎 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑏 ∈ 𝑎 ({𝑏} × 𝒫 𝑏)))
18		dmeq 5883	. . . . . . . 8 ⊢ (𝑐 = 𝑎 → dom 𝑐 = dom 𝑎)
19	18	pweqd 4592	. . . . . . 7 ⊢ (𝑐 = 𝑎 → 𝒫 dom 𝑐 = 𝒫 dom 𝑎)
20		imaeq1 6042	. . . . . . . 8 ⊢ (𝑐 = 𝑎 → (𝑐 “ 𝑑) = (𝑎 “ 𝑑))
21	20	fveq2d 6880	. . . . . . 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 5207	. . . . . 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 6043	. . . . . . . 8 ⊢ (𝑑 = 𝑏 → (𝑎 “ 𝑑) = (𝑎 “ 𝑏))
24	23	fveq2d 6880	. . . . . . 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 5225	. . . . . 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 2786	. . . . 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 5225	. . . 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 2735	. . . 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 10256	. . 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 6889	. . . . 5 ⊢ (𝑅1‘ω) ∈ V
31	9, 30	eqeltrri 2831	. . . 4 ⊢ ∪ (𝑅1 “ ω) ∈ V
32	31	f1oen 8987	. . 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 5140	1 ⊢ (𝑅1‘ω) ≈ ω

Syntax hints:   = wceq 1540   ∈ wcel 2108  Vcvv 3459   ∩ cin 3925  ∅c0 4308  𝒫 cpw 4575  {csn 4601  ∪ cuni 4883  ∪ ciun 4967   class class class wbr 5119   ↦ cmpt 5201   × cxp 5652  dom cdm 5654   “ cima 5657  Oncon0 6352  Lim wlim 6353  Fun wfun 6525   Fn wfn 6526  –1-1-onto→wf1o 6530  ‘cfv 6531  ωcom 7861  reccrdg 8423   ≈ cen 8956  Fincfn 8959  𝑅₁cr1 9776  cardccrd 9949

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-inf2 9655

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-er 8719  df-map 8842  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-r1 9778  df-rank 9779  df-dju 9915  df-card 9953

This theorem is referenced by: (None)