Theorem fin23lem22 10306

Description: Lemma for fin23 10368 but could be used elsewhere if we find a good name for it. Explicit construction of a bijection (actually an isomorphism, see fin23lem27 10307) 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 10305	. . 3 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∃!𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖)
3		riotacl 7384	. . 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 778	. . . 4 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑎 ∈ 𝑆) → 𝑆 ⊆ ω)
6		simpr 489	. . . 4 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ 𝑆)
7	5, 6	sseldd 3938	. . 3 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ ω)
8		nnfi 9148	. . 3 ⊢ (𝑎 ∈ ω → 𝑎 ∈ Fin)
9		infi 9226	. . 3 ⊢ (𝑎 ∈ Fin → (𝑎 ∩ 𝑆) ∈ Fin)
10		ficardom 9943	. . 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 9945	. . . . . . 7 ⊢ (𝑖 ∈ ω → (card‘𝑖) = 𝑖)
13	12	eqcomd 2769	. . . . . 6 ⊢ (𝑖 ∈ ω → 𝑖 = (card‘𝑖))
14	13	eqeq1d 2765	. . . . 5 ⊢ (𝑖 ∈ ω → (𝑖 = (card‘(𝑎 ∩ 𝑆)) ↔ (card‘𝑖) = (card‘(𝑎 ∩ 𝑆))))
15		eqcom 2770	. . . . 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 740	. . 3 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → (𝑖 = (card‘(𝑎 ∩ 𝑆)) ↔ (card‘(𝑎 ∩ 𝑆)) = (card‘𝑖)))
18		simpll 778	. . . . . . 7 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → 𝑆 ⊆ ω)
19		simprr 784	. . . . . . 7 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → 𝑎 ∈ 𝑆)
20	18, 19	sseldd 3938	. . . . . 6 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → 𝑎 ∈ ω)
21		nnon 7864	. . . . . 6 ⊢ (𝑎 ∈ ω → 𝑎 ∈ On)
22		onenon 9931	. . . . . 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 4189	. . . . 5 ⊢ (𝑎 ∩ 𝑆) ⊆ 𝑎
25		ssnum 10019	. . . . 5 ⊢ ((𝑎 ∈ dom card ∧ (𝑎 ∩ 𝑆) ⊆ 𝑎) → (𝑎 ∩ 𝑆) ∈ dom card)
26	23, 24, 25	sylancl 597	. . . 4 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → (𝑎 ∩ 𝑆) ∈ dom card)
27		nnon 7864	. . . . . 6 ⊢ (𝑖 ∈ ω → 𝑖 ∈ On)
28	27	ad2antrl 740	. . . . 5 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → 𝑖 ∈ On)
29		onenon 9931	. . . . 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 9969	. . . 4 ⊢ (((𝑎 ∩ 𝑆) ∈ dom card ∧ 𝑖 ∈ dom card) → ((card‘(𝑎 ∩ 𝑆)) = (card‘𝑖) ↔ (𝑎 ∩ 𝑆) ≈ 𝑖))
32	26, 30, 31	syl2anc 595	. . 3 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → ((card‘(𝑎 ∩ 𝑆)) = (card‘𝑖) ↔ (𝑎 ∩ 𝑆) ≈ 𝑖))
33	2	adantrr 729	. . . . 5 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → ∃!𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖)
34		ineq1 4166	. . . . . . 7 ⊢ (𝑗 = 𝑎 → (𝑗 ∩ 𝑆) = (𝑎 ∩ 𝑆))
35	34	breq1d 5119	. . . . . 6 ⊢ (𝑗 = 𝑎 → ((𝑗 ∩ 𝑆) ≈ 𝑖 ↔ (𝑎 ∩ 𝑆) ≈ 𝑖))
36	35	riota2 7392	. . . . 5 ⊢ ((𝑎 ∈ 𝑆 ∧ ∃!𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖) → ((𝑎 ∩ 𝑆) ≈ 𝑖 ↔ (℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖) = 𝑎))
37	19, 33, 36	syl2anc 595	. . . 4 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → ((𝑎 ∩ 𝑆) ≈ 𝑖 ↔ (℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖) = 𝑎))
38		eqcom 2770	. . . 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 7664	1 ⊢ ((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) → 𝐶:ω–1-1-onto→𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∃!wreu 3367   ∩ cin 3904   ⊆ wss 3905   class class class wbr 5109   ↦ cmpt 5192  dom cdm 5661  Oncon0 6360  –1-1-onto→wf1o 6535  ‘cfv 6536  ℩crio 7366  ωcom 7858   ≈ cen 8936  Fincfn 8939  cardccrd 9917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-1o 8449  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9921

This theorem is referenced by:  fin23lem27  10307  fin23lem28  10319  fin23lem30  10321  isf32lem6  10337  isf32lem7  10338  isf32lem8  10339