Theorem r1om 9319

Description: The set of hereditarily finite sets is countable. See ackbij2 9318 for an explicit bijection that works without Infinity. See also r1omALT 9851. (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 8755	. . . 4 ⊢ ω ∈ V
2		limom 7278	. . . 4 ⊢ Lim ω
3		r1lim 8850	. . . 4 ⊢ ((ω ∈ V ∧ Lim ω) → (𝑅1‘ω) = ∪ 𝑎 ∈ ω (𝑅1‘𝑎))
4	1, 2, 3	mp2an 683	. . 3 ⊢ (𝑅1‘ω) = ∪ 𝑎 ∈ ω (𝑅1‘𝑎)
5		r1fnon 8845	. . . 4 ⊢ 𝑅1 Fn On
6		fnfun 6166	. . . 4 ⊢ (𝑅1 Fn On → Fun 𝑅1)
7		funiunfv 6698	. . . 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 2787	. 2 ⊢ (𝑅1‘ω) = ∪ (𝑅1 “ ω)
10		iuneq1 4690	. . . . . . 7 ⊢ (𝑒 = 𝑎 → ∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓) = ∪ 𝑓 ∈ 𝑎 ({𝑓} × 𝒫 𝑓))
11		sneq 4344	. . . . . . . . 9 ⊢ (𝑓 = 𝑏 → {𝑓} = {𝑏})
12		pweq 4318	. . . . . . . . 9 ⊢ (𝑓 = 𝑏 → 𝒫 𝑓 = 𝒫 𝑏)
13	11, 12	xpeq12d 5308	. . . . . . . 8 ⊢ (𝑓 = 𝑏 → ({𝑓} × 𝒫 𝑓) = ({𝑏} × 𝒫 𝑏))
14	13	cbviunv 4715	. . . . . . 7 ⊢ ∪ 𝑓 ∈ 𝑎 ({𝑓} × 𝒫 𝑓) = ∪ 𝑏 ∈ 𝑎 ({𝑏} × 𝒫 𝑏)
15	10, 14	syl6eq 2815	. . . . . 6 ⊢ (𝑒 = 𝑎 → ∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓) = ∪ 𝑏 ∈ 𝑎 ({𝑏} × 𝒫 𝑏))
16	15	fveq2d 6379	. . . . 5 ⊢ (𝑒 = 𝑎 → (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓)) = (card‘∪ 𝑏 ∈ 𝑎 ({𝑏} × 𝒫 𝑏)))
17	16	cbvmptv 4909	. . . 4 ⊢ (𝑒 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑒 ({𝑓} × 𝒫 𝑓))) = (𝑎 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑏 ∈ 𝑎 ({𝑏} × 𝒫 𝑏)))
18		dmeq 5492	. . . . . . . 8 ⊢ (𝑐 = 𝑎 → dom 𝑐 = dom 𝑎)
19	18	pweqd 4320	. . . . . . 7 ⊢ (𝑐 = 𝑎 → 𝒫 dom 𝑐 = 𝒫 dom 𝑎)
20		imaeq1 5643	. . . . . . . 8 ⊢ (𝑐 = 𝑎 → (𝑐 “ 𝑑) = (𝑎 “ 𝑑))
21	20	fveq2d 6379	. . . . . . 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 4892	. . . . . 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 5644	. . . . . . . 8 ⊢ (𝑑 = 𝑏 → (𝑎 “ 𝑑) = (𝑎 “ 𝑏))
24	23	fveq2d 6379	. . . . . . 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 4909	. . . . . 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	syl6eq 2815	. . . . 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 4909	. . . 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 2765	. . . 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 9318	. . 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 6388	. . . . 5 ⊢ (𝑅1‘ω) ∈ V
31	9, 30	eqeltrri 2841	. . . 4 ⊢ ∪ (𝑅1 “ ω) ∈ V
32	31	f1oen 8181	. . 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 4830	1 ⊢ (𝑅1‘ω) ≈ ω

Syntax hints:   = wceq 1652   ∈ wcel 2155  Vcvv 3350   ∩ cin 3731  ∅c0 4079  𝒫 cpw 4315  {csn 4334  ∪ cuni 4594  ∪ ciun 4676   class class class wbr 4809   ↦ cmpt 4888   × cxp 5275  dom cdm 5277   “ cima 5280  Oncon0 5908  Lim wlim 5909  Fun wfun 6062   Fn wfn 6063  –1-1-onto→wf1o 6067  ‘cfv 6068  ωcom 7263  reccrdg 7709   ≈ cen 8157  Fincfn 8160  𝑅₁cr1 8840  cardccrd 9012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-map 8062  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-r1 8842  df-rank 8843  df-card 9016  df-cda 9243

This theorem is referenced by: (None)