Theorem fin23lem22 10284

Description: Lemma for fin23 10346 but could be used elsewhere if we find a good name for it. Explicit construction of a bijection (actually an isomorphism, see fin23lem27 10285) 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 10283	. . 3 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∃!𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖)
3		riotacl 7370	. . 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 776	. . . 4 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑎 ∈ 𝑆) → 𝑆 ⊆ ω)
6		simpr 488	. . . 4 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ 𝑆)
7	5, 6	sseldd 3937	. . 3 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑎 ∈ 𝑆) → 𝑎 ∈ ω)
8		nnfi 9136	. . 3 ⊢ (𝑎 ∈ ω → 𝑎 ∈ Fin)
9		infi 9214	. . 3 ⊢ (𝑎 ∈ Fin → (𝑎 ∩ 𝑆) ∈ Fin)
10		ficardom 9919	. . 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 9921	. . . . . . 7 ⊢ (𝑖 ∈ ω → (card‘𝑖) = 𝑖)
13	12	eqcomd 2768	. . . . . 6 ⊢ (𝑖 ∈ ω → 𝑖 = (card‘𝑖))
14	13	eqeq1d 2764	. . . . 5 ⊢ (𝑖 ∈ ω → (𝑖 = (card‘(𝑎 ∩ 𝑆)) ↔ (card‘𝑖) = (card‘(𝑎 ∩ 𝑆))))
15		eqcom 2769	. . . . 5 ⊢ ((card‘𝑖) = (card‘(𝑎 ∩ 𝑆)) ↔ (card‘(𝑎 ∩ 𝑆)) = (card‘𝑖))
16	14, 15	bitrdi 289	. . . 4 ⊢ (𝑖 ∈ ω → (𝑖 = (card‘(𝑎 ∩ 𝑆)) ↔ (card‘(𝑎 ∩ 𝑆)) = (card‘𝑖)))
17	16	ad2antrl 738	. . 3 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → (𝑖 = (card‘(𝑎 ∩ 𝑆)) ↔ (card‘(𝑎 ∩ 𝑆)) = (card‘𝑖)))
18		simpll 776	. . . . . . 7 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → 𝑆 ⊆ ω)
19		simprr 782	. . . . . . 7 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → 𝑎 ∈ 𝑆)
20	18, 19	sseldd 3937	. . . . . 6 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → 𝑎 ∈ ω)
21		nnon 7852	. . . . . 6 ⊢ (𝑎 ∈ ω → 𝑎 ∈ On)
22		onenon 9907	. . . . . 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 4188	. . . . 5 ⊢ (𝑎 ∩ 𝑆) ⊆ 𝑎
25		ssnum 9995	. . . . 5 ⊢ ((𝑎 ∈ dom card ∧ (𝑎 ∩ 𝑆) ⊆ 𝑎) → (𝑎 ∩ 𝑆) ∈ dom card)
26	23, 24, 25	sylancl 595	. . . 4 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → (𝑎 ∩ 𝑆) ∈ dom card)
27		nnon 7852	. . . . . 6 ⊢ (𝑖 ∈ ω → 𝑖 ∈ On)
28	27	ad2antrl 738	. . . . 5 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → 𝑖 ∈ On)
29		onenon 9907	. . . . 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 9945	. . . 4 ⊢ (((𝑎 ∩ 𝑆) ∈ dom card ∧ 𝑖 ∈ dom card) → ((card‘(𝑎 ∩ 𝑆)) = (card‘𝑖) ↔ (𝑎 ∩ 𝑆) ≈ 𝑖))
32	26, 30, 31	syl2anc 593	. . 3 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → ((card‘(𝑎 ∩ 𝑆)) = (card‘𝑖) ↔ (𝑎 ∩ 𝑆) ≈ 𝑖))
33	2	adantrr 727	. . . . 5 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → ∃!𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖)
34		ineq1 4165	. . . . . . 7 ⊢ (𝑗 = 𝑎 → (𝑗 ∩ 𝑆) = (𝑎 ∩ 𝑆))
35	34	breq1d 5110	. . . . . 6 ⊢ (𝑗 = 𝑎 → ((𝑗 ∩ 𝑆) ≈ 𝑖 ↔ (𝑎 ∩ 𝑆) ≈ 𝑖))
36	35	riota2 7378	. . . . 5 ⊢ ((𝑎 ∈ 𝑆 ∧ ∃!𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖) → ((𝑎 ∩ 𝑆) ≈ 𝑖 ↔ (℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖) = 𝑎))
37	19, 33, 36	syl2anc 593	. . . 4 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → ((𝑎 ∩ 𝑆) ≈ 𝑖 ↔ (℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖) = 𝑎))
38		eqcom 2769	. . . 4 ⊢ ((℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖) = 𝑎 ↔ 𝑎 = (℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖))
39	37, 38	bitrdi 289	. . 3 ⊢ (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎 ∈ 𝑆)) → ((𝑎 ∩ 𝑆) ≈ 𝑖 ↔ 𝑎 = (℩𝑗 ∈ 𝑆 (𝑗 ∩ 𝑆) ≈ 𝑖)))
40	17, 32, 39	3bitrd 307	. 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 7650	1 ⊢ ((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) → 𝐶:ω–1-1-onto→𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∃!wreu 3365   ∩ cin 3903   ⊆ wss 3904   class class class wbr 5100   ↦ cmpt 5181  dom cdm 5647  Oncon0 6346  –1-1-onto→wf1o 6520  ‘cfv 6521  ℩crio 7352  ωcom 7846   ≈ cen 8924  Fincfn 8927  cardccrd 9893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-isom 6530  df-riota 7353  df-ov 7399  df-om 7847  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-1o 8437  df-er 8678  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-card 9897

This theorem is referenced by:  fin23lem27  10285  fin23lem28  10297  fin23lem30  10299  isf32lem6  10315  isf32lem7  10316  isf32lem8  10317