Theorem fin23lem22 10330

Description: Lemma for fin23 10392 but could be used elsewhere if we find a good name for it. Explicit construction of a bijection (actually an isomorphism, see fin23lem27 10331) between an infinite subset of ω and ω itself. (Contributed by Stefan O'Rear, 1-Nov-2014.)

Hypothesis

Ref	Expression
fin23lem22.b	⊢ 𝐶 = (𝑖 ∈ ω ↦ (℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖))

Assertion

Ref	Expression
fin23lem22	⊢ ((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) → 𝐶:ω–1-1-onto→𝑆)

Distinct variable group:   𝑖,𝑗,𝑆

Allowed substitution hints:   𝐶(𝑖, 𝑗)

Proof of Theorem fin23lem22

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fin23lem22.b	. 2 ⊢ 𝐶 = (𝑖 ∈ ω ↦ (℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖))
2		fin23lem23 10329	. . 3 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∃!𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖)
3		riotacl 7388	. . 3 ⊢ (∃!𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖 → (℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖) ∈ 𝑆)
4	2, 3	syl 18	. 2 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → (℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖) ∈ 𝑆)
5		simpll 779	. . . 4 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑎 ∈ 𝑆) → 𝑆 ⊆ ω)
6		simpr 490	. . . 4 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ 𝑆)
7	5, 6	sseldd 3932	. . 3 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ ω)
8		nnfi 9163	. . 3 ⊢ (𝑎 ∈ ω → 𝑎 ∈ Fin)
9		infi 9241	. . 3 ⊢ (𝑎 ∈ Fin → (𝑎 ∩ 𝑆) ∈ Fin)
10		ficardom 9967	. . 3 ⊢ ((𝑎 ∩ 𝑆) ∈ Fin → (card‘(𝑎 ∩ 𝑆)) ∈ ω)
11	7, 8, 9, 10	4syl 20	. 2 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑎 ∈ 𝑆) → (card‘(𝑎 ∩ 𝑆)) ∈ ω)
12		cardnn 9969	. . . . . . 7 ⊢ (𝑖 ∈ ω → (card‘𝑖) = 𝑖)
13	12	eqcomd 2766	. . . . . 6 ⊢ (𝑖 ∈ ω → 𝑖 = (card‘𝑖))
14	13	eqeq1d 2762	. . . . 5 ⊢ (𝑖 ∈ ω → (𝑖 = (card‘(𝑎 ∩ 𝑆)) ↔ (card‘𝑖) = (card‘(𝑎 ∩ 𝑆))))
15		eqcom 2767	. . . . 5 ⊢ ((card‘𝑖) = (card‘(𝑎 ∩ 𝑆)) ↔ (card‘(𝑎 ∩ 𝑆)) = (card‘𝑖))
16	14, 15	bitrdi 290	. . . 4 ⊢ (𝑖 ∈ ω → (𝑖 = (card‘(𝑎 ∩ 𝑆)) ↔ (card‘(𝑎 ∩ 𝑆)) = (card‘𝑖)))
17	16	ad2antrl 741	. . 3 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → (𝑖 = (card‘(𝑎 ∩ 𝑆)) ↔ (card‘(𝑎 ∩ 𝑆)) = (card‘𝑖)))
18		simpll 779	. . . . . . 7 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → 𝑆 ⊆ ω)
19		simprr 785	. . . . . . 7 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → 𝑎 ∈ 𝑆)
20	18, 19	sseldd 3932	. . . . . 6 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → 𝑎 ∈ ω)
21		nnon 7869	. . . . . 6 ⊢ (𝑎 ∈ ω → 𝑎 ∈ On)
22		onenon 9955	. . . . . 6 ⊢ (𝑎 ∈ On → 𝑎 ∈ dom card)
23	20, 21, 22	3syl 19	. . . . 5 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → 𝑎 ∈ dom card)
24		inss1 4182	. . . . 5 ⊢ (𝑎 ∩ 𝑆) ⊆ 𝑎
25		ssnum 10043	. . . . 5 ⊢ ((𝑎 ∈ dom card ∧ (𝑎 ∩ 𝑆) ⊆ 𝑎) → (𝑎 ∩ 𝑆) ∈ dom card)
26	23, 24, 25	sylancl 598	. . . 4 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → (𝑎 ∩ 𝑆) ∈ dom card)
27		nnon 7869	. . . . . 6 ⊢ (𝑖 ∈ ω → 𝑖 ∈ On)
28	27	ad2antrl 741	. . . . 5 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → 𝑖 ∈ On)
29		onenon 9955	. . . . 5 ⊢ (𝑖 ∈ On → 𝑖 ∈ dom card)
30	28, 29	syl 18	. . . 4 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → 𝑖 ∈ dom card)
31		carden2 9993	. . . 4 ⊢ (((𝑎 ∩ 𝑆) ∈ dom card ∧ 𝑖 ∈ dom card) → ((card‘(𝑎 ∩ 𝑆)) = (card‘𝑖) ↔ (𝑎 ∩ 𝑆) ≈ 𝑖))
32	26, 30, 31	syl2anc 596	. . 3 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → ((card‘(𝑎 ∩ 𝑆)) = (card‘𝑖) ↔ (𝑎 ∩ 𝑆) ≈ 𝑖))
33	2	adantrr 730	. . . . 5 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → ∃!𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖)
34		ineq1 4159	. . . . . . 7 ⊢ (𝑗 = 𝑎 → (𝑗 ∩ 𝑆) = (𝑎 ∩ 𝑆))
35	34	breq1d 5113	. . . . . 6 ⊢ (𝑗 = 𝑎 → ((𝑗 ∩ 𝑆) ≈ 𝑖 ↔ (𝑎 ∩ 𝑆) ≈ 𝑖))
36	35	riota2 7396	. . . . 5 ⊢ ((𝑎 ∈ 𝑆 ∧ ∃!𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖) → ((𝑎 ∩ 𝑆) ≈ 𝑖 ↔ (℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖) = 𝑎))
37	19, 33, 36	syl2anc 596	. . . 4 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → ((𝑎 ∩ 𝑆) ≈ 𝑖 ↔ (℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖) = 𝑎))
38		eqcom 2767	. . . 4 ⊢ ((℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖) = 𝑎 ↔ 𝑎 = (℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖))
39	37, 38	bitrdi 290	. . 3 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → ((𝑎 ∩ 𝑆) ≈ 𝑖 ↔ 𝑎 = (℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖)))
40	17, 32, 39	3bitrd 308	. 2 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → (𝑖 = (card‘(𝑎 ∩ 𝑆)) ↔ 𝑎 = (℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖)))
41	1, 4, 11, 40	f1o2d 7669	1 ⊢ ((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) → 𝐶:ω–1-1-onto→𝑆)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃!wreu 3363   ∩ cin 3898   ⊆ wss 3899   class class class wbr 5103   ↦ cmpt 5186  dom cdm 5655  Oncon0 6357  –1-1-onto→wf1o 6532  ‘cfv 6533  ℩crio 7370  ωcom 7863   ≈ cen 8950  Fincfn 8953  cardccrd 9941

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7371  df-ov 7417  df-om 7864  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-1o 8456  df-er 8697  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-card 9945

This theorem is used by:  fin23lem27  10331  fin23lem28  10343  fin23lem30  10345  isf32lem6  10361  isf32lem7  10362  isf32lem8  10363