Theorem fin23lem22 10243

Description: Lemma for fin23 10305 but could be used elsewhere if we find a good name for it. Explicit construction of a bijection (actually an isomorphism, see fin23lem27 10244) 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 10242	. . 3 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∃!𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖)
3		riotacl 7335	. . 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 767	. . . 4 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑎 ∈ 𝑆) → 𝑆 ⊆ ω)
6		simpr 484	. . . 4 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ 𝑆)
7	5, 6	sseldd 3923	. . 3 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ ω)
8		nnfi 9096	. . 3 ⊢ (𝑎 ∈ ω → 𝑎 ∈ Fin)
9		infi 9174	. . 3 ⊢ (𝑎 ∈ Fin → (𝑎 ∩ 𝑆) ∈ Fin)
10		ficardom 9879	. . 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 9881	. . . . . . 7 ⊢ (𝑖 ∈ ω → (card‘𝑖) = 𝑖)
13	12	eqcomd 2743	. . . . . 6 ⊢ (𝑖 ∈ ω → 𝑖 = (card‘𝑖))
14	13	eqeq1d 2739	. . . . 5 ⊢ (𝑖 ∈ ω → (𝑖 = (card‘(𝑎 ∩ 𝑆)) ↔ (card‘𝑖) = (card‘(𝑎 ∩ 𝑆))))
15		eqcom 2744	. . . . 5 ⊢ ((card‘𝑖) = (card‘(𝑎 ∩ 𝑆)) ↔ (card‘(𝑎 ∩ 𝑆)) = (card‘𝑖))
16	14, 15	bitrdi 287	. . . 4 ⊢ (𝑖 ∈ ω → (𝑖 = (card‘(𝑎 ∩ 𝑆)) ↔ (card‘(𝑎 ∩ 𝑆)) = (card‘𝑖)))
17	16	ad2antrl 729	. . 3 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → (𝑖 = (card‘(𝑎 ∩ 𝑆)) ↔ (card‘(𝑎 ∩ 𝑆)) = (card‘𝑖)))
18		simpll 767	. . . . . . 7 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → 𝑆 ⊆ ω)
19		simprr 773	. . . . . . 7 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → 𝑎 ∈ 𝑆)
20	18, 19	sseldd 3923	. . . . . 6 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → 𝑎 ∈ ω)
21		nnon 7817	. . . . . 6 ⊢ (𝑎 ∈ ω → 𝑎 ∈ On)
22		onenon 9867	. . . . . 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 4178	. . . . 5 ⊢ (𝑎 ∩ 𝑆) ⊆ 𝑎
25		ssnum 9955	. . . . 5 ⊢ ((𝑎 ∈ dom card ∧ (𝑎 ∩ 𝑆) ⊆ 𝑎) → (𝑎 ∩ 𝑆) ∈ dom card)
26	23, 24, 25	sylancl 587	. . . 4 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → (𝑎 ∩ 𝑆) ∈ dom card)
27		nnon 7817	. . . . . 6 ⊢ (𝑖 ∈ ω → 𝑖 ∈ On)
28	27	ad2antrl 729	. . . . 5 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → 𝑖 ∈ On)
29		onenon 9867	. . . . 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 9905	. . . 4 ⊢ (((𝑎 ∩ 𝑆) ∈ dom card ∧ 𝑖 ∈ dom card) → ((card‘(𝑎 ∩ 𝑆)) = (card‘𝑖) ↔ (𝑎 ∩ 𝑆) ≈ 𝑖))
32	26, 30, 31	syl2anc 585	. . 3 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → ((card‘(𝑎 ∩ 𝑆)) = (card‘𝑖) ↔ (𝑎 ∩ 𝑆) ≈ 𝑖))
33	2	adantrr 718	. . . . 5 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → ∃!𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖)
34		ineq1 4154	. . . . . . 7 ⊢ (𝑗 = 𝑎 → (𝑗 ∩ 𝑆) = (𝑎 ∩ 𝑆))
35	34	breq1d 5096	. . . . . 6 ⊢ (𝑗 = 𝑎 → ((𝑗 ∩ 𝑆) ≈ 𝑖 ↔ (𝑎 ∩ 𝑆) ≈ 𝑖))
36	35	riota2 7343	. . . . 5 ⊢ ((𝑎 ∈ 𝑆 ∧ ∃!𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖) → ((𝑎 ∩ 𝑆) ≈ 𝑖 ↔ (℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖) = 𝑎))
37	19, 33, 36	syl2anc 585	. . . 4 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → ((𝑎 ∩ 𝑆) ≈ 𝑖 ↔ (℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖) = 𝑎))
38		eqcom 2744	. . . 4 ⊢ ((℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖) = 𝑎 ↔ 𝑎 = (℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖))
39	37, 38	bitrdi 287	. . 3 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → ((𝑎 ∩ 𝑆) ≈ 𝑖 ↔ 𝑎 = (℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖)))
40	17, 32, 39	3bitrd 305	. 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 7615	1 ⊢ ((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) → 𝐶:ω–1-1-onto→𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃!wreu 3341   ∩ cin 3889   ⊆ wss 3890   class class class wbr 5086   ↦ cmpt 5167  dom cdm 5625  Oncon0 6318  –1-1-onto→wf1o 6492  ‘cfv 6493  ℩crio 7317  ωcom 7811   ≈ cen 8884  Fincfn 8887  cardccrd 9853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7318  df-ov 7364  df-om 7812  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-1o 8399  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-card 9857

This theorem is referenced by:  fin23lem27  10244  fin23lem28  10256  fin23lem30  10258  isf32lem6  10274  isf32lem7  10275  isf32lem8  10276