Theorem fin23lem22 9032

Description: Lemma for fin23 9094 but could be used elsewhere if we find a good name for it. Explicit construction of a bijection (actually an isomorphism, see fin23lem27 9033) 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 9031	. . 3 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∃!𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖)
3		riotacl 6525	. . 3 ⊢ (∃!𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖 → (℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖) ∈ 𝑆)
4	2, 3	syl 17	. 2 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → (℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖) ∈ 𝑆)
5		simpll 786	. . . 4 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑎 ∈ 𝑆) → 𝑆 ⊆ ω)
6		simpr 476	. . . 4 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ 𝑆)
7	5, 6	sseldd 3569	. . 3 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ ω)
8		nnfi 8038	. . 3 ⊢ (𝑎 ∈ ω → 𝑎 ∈ Fin)
9		infi 8069	. . 3 ⊢ (𝑎 ∈ Fin → (𝑎 ∩ 𝑆) ∈ Fin)
10		ficardom 8670	. . 3 ⊢ ((𝑎 ∩ 𝑆) ∈ Fin → (card‘(𝑎 ∩ 𝑆)) ∈ ω)
11	7, 8, 9, 10	4syl 19	. 2 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑎 ∈ 𝑆) → (card‘(𝑎 ∩ 𝑆)) ∈ ω)
12		cardnn 8672	. . . . . . 7 ⊢ (𝑖 ∈ ω → (card‘𝑖) = 𝑖)
13	12	eqcomd 2616	. . . . . 6 ⊢ (𝑖 ∈ ω → 𝑖 = (card‘𝑖))
14	13	eqeq1d 2612	. . . . 5 ⊢ (𝑖 ∈ ω → (𝑖 = (card‘(𝑎 ∩ 𝑆)) ↔ (card‘𝑖) = (card‘(𝑎 ∩ 𝑆))))
15		eqcom 2617	. . . . 5 ⊢ ((card‘𝑖) = (card‘(𝑎 ∩ 𝑆)) ↔ (card‘(𝑎 ∩ 𝑆)) = (card‘𝑖))
16	14, 15	syl6bb 275	. . . 4 ⊢ (𝑖 ∈ ω → (𝑖 = (card‘(𝑎 ∩ 𝑆)) ↔ (card‘(𝑎 ∩ 𝑆)) = (card‘𝑖)))
17	16	ad2antrl 760	. . 3 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → (𝑖 = (card‘(𝑎 ∩ 𝑆)) ↔ (card‘(𝑎 ∩ 𝑆)) = (card‘𝑖)))
18		simpll 786	. . . . . . 7 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → 𝑆 ⊆ ω)
19		simprr 792	. . . . . . 7 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → 𝑎 ∈ 𝑆)
20	18, 19	sseldd 3569	. . . . . 6 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → 𝑎 ∈ ω)
21		nnon 6963	. . . . . 6 ⊢ (𝑎 ∈ ω → 𝑎 ∈ On)
22		onenon 8658	. . . . . 6 ⊢ (𝑎 ∈ On → 𝑎 ∈ dom card)
23	20, 21, 22	3syl 18	. . . . 5 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → 𝑎 ∈ dom card)
24		inss1 3795	. . . . 5 ⊢ (𝑎 ∩ 𝑆) ⊆ 𝑎
25		ssnum 8745	. . . . 5 ⊢ ((𝑎 ∈ dom card ∧ (𝑎 ∩ 𝑆) ⊆ 𝑎) → (𝑎 ∩ 𝑆) ∈ dom card)
26	23, 24, 25	sylancl 693	. . . 4 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → (𝑎 ∩ 𝑆) ∈ dom card)
27		nnon 6963	. . . . . 6 ⊢ (𝑖 ∈ ω → 𝑖 ∈ On)
28	27	ad2antrl 760	. . . . 5 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → 𝑖 ∈ On)
29		onenon 8658	. . . . 5 ⊢ (𝑖 ∈ On → 𝑖 ∈ dom card)
30	28, 29	syl 17	. . . 4 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → 𝑖 ∈ dom card)
31		carden2 8696	. . . 4 ⊢ (((𝑎 ∩ 𝑆) ∈ dom card ∧ 𝑖 ∈ dom card) → ((card‘(𝑎 ∩ 𝑆)) = (card‘𝑖) ↔ (𝑎 ∩ 𝑆) ≈ 𝑖))
32	26, 30, 31	syl2anc 691	. . 3 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → ((card‘(𝑎 ∩ 𝑆)) = (card‘𝑖) ↔ (𝑎 ∩ 𝑆) ≈ 𝑖))
33	2	adantrr 749	. . . . 5 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → ∃!𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖)
34		ineq1 3769	. . . . . . 7 ⊢ (𝑗 = 𝑎 → (𝑗 ∩ 𝑆) = (𝑎 ∩ 𝑆))
35	34	breq1d 4593	. . . . . 6 ⊢ (𝑗 = 𝑎 → ((𝑗 ∩ 𝑆) ≈ 𝑖 ↔ (𝑎 ∩ 𝑆) ≈ 𝑖))
36	35	riota2 6533	. . . . 5 ⊢ ((𝑎 ∈ 𝑆 ∧ ∃!𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖) → ((𝑎 ∩ 𝑆) ≈ 𝑖 ↔ (℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖) = 𝑎))
37	19, 33, 36	syl2anc 691	. . . 4 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → ((𝑎 ∩ 𝑆) ≈ 𝑖 ↔ (℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖) = 𝑎))
38		eqcom 2617	. . . 4 ⊢ ((℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖) = 𝑎 ↔ 𝑎 = (℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖))
39	37, 38	syl6bb 275	. . 3 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → ((𝑎 ∩ 𝑆) ≈ 𝑖 ↔ 𝑎 = (℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖)))
40	17, 32, 39	3bitrd 293	. 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 6785	1 ⊢ ((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) → 𝐶:ω–1-1-onto→𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃!wreu 2898   ∩ cin 3539   ⊆ wss 3540   class class class wbr 4583   ↦ cmpt 4643  dom cdm 5038  Oncon0 5640  –1-1-onto→wf1o 5803  ‘cfv 5804  ℩crio 6510  ωcom 6957   ≈ cen 7838  Fincfn 7841  cardccrd 8644

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-om 6958  df-wrecs 7294  df-recs 7355  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648

This theorem is referenced by:  fin23lem27  9033  fin23lem28  9045  fin23lem30  9047  isf32lem6  9063  isf32lem7  9064  isf32lem8  9065