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Theorem fin23lem22 9742
Description: Lemma for fin23 9804 but could be used elsewhere if we find a good name for it. Explicit construction of a bijection (actually an isomorphism, see fin23lem27 9743) 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 ↦ (𝑗𝑆 (𝑗𝑆) ≈ 𝑖))
Assertion
Ref Expression
fin23lem22 ((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) → 𝐶:ω–1-1-onto𝑆)
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.
StepHypRef Expression
1 fin23lem22.b . 2 𝐶 = (𝑖 ∈ ω ↦ (𝑗𝑆 (𝑗𝑆) ≈ 𝑖))
2 fin23lem23 9741 . . 3 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∃!𝑗𝑆 (𝑗𝑆) ≈ 𝑖)
3 riotacl 7114 . . 3 (∃!𝑗𝑆 (𝑗𝑆) ≈ 𝑖 → (𝑗𝑆 (𝑗𝑆) ≈ 𝑖) ∈ 𝑆)
42, 3syl 17 . 2 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → (𝑗𝑆 (𝑗𝑆) ≈ 𝑖) ∈ 𝑆)
5 simpll 766 . . . 4 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑎𝑆) → 𝑆 ⊆ ω)
6 simpr 488 . . . 4 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑎𝑆) → 𝑎𝑆)
75, 6sseldd 3919 . . 3 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑎𝑆) → 𝑎 ∈ ω)
8 nnfi 8700 . . 3 (𝑎 ∈ ω → 𝑎 ∈ Fin)
9 infi 8730 . . 3 (𝑎 ∈ Fin → (𝑎𝑆) ∈ Fin)
10 ficardom 9378 . . 3 ((𝑎𝑆) ∈ Fin → (card‘(𝑎𝑆)) ∈ ω)
117, 8, 9, 104syl 19 . 2 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑎𝑆) → (card‘(𝑎𝑆)) ∈ ω)
12 cardnn 9380 . . . . . . 7 (𝑖 ∈ ω → (card‘𝑖) = 𝑖)
1312eqcomd 2807 . . . . . 6 (𝑖 ∈ ω → 𝑖 = (card‘𝑖))
1413eqeq1d 2803 . . . . 5 (𝑖 ∈ ω → (𝑖 = (card‘(𝑎𝑆)) ↔ (card‘𝑖) = (card‘(𝑎𝑆))))
15 eqcom 2808 . . . . 5 ((card‘𝑖) = (card‘(𝑎𝑆)) ↔ (card‘(𝑎𝑆)) = (card‘𝑖))
1614, 15syl6bb 290 . . . 4 (𝑖 ∈ ω → (𝑖 = (card‘(𝑎𝑆)) ↔ (card‘(𝑎𝑆)) = (card‘𝑖)))
1716ad2antrl 727 . . 3 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → (𝑖 = (card‘(𝑎𝑆)) ↔ (card‘(𝑎𝑆)) = (card‘𝑖)))
18 simpll 766 . . . . . . 7 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → 𝑆 ⊆ ω)
19 simprr 772 . . . . . . 7 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → 𝑎𝑆)
2018, 19sseldd 3919 . . . . . 6 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → 𝑎 ∈ ω)
21 nnon 7570 . . . . . 6 (𝑎 ∈ ω → 𝑎 ∈ On)
22 onenon 9366 . . . . . 6 (𝑎 ∈ On → 𝑎 ∈ dom card)
2320, 21, 223syl 18 . . . . 5 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → 𝑎 ∈ dom card)
24 inss1 4158 . . . . 5 (𝑎𝑆) ⊆ 𝑎
25 ssnum 9454 . . . . 5 ((𝑎 ∈ dom card ∧ (𝑎𝑆) ⊆ 𝑎) → (𝑎𝑆) ∈ dom card)
2623, 24, 25sylancl 589 . . . 4 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → (𝑎𝑆) ∈ dom card)
27 nnon 7570 . . . . . 6 (𝑖 ∈ ω → 𝑖 ∈ On)
2827ad2antrl 727 . . . . 5 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → 𝑖 ∈ On)
29 onenon 9366 . . . . 5 (𝑖 ∈ On → 𝑖 ∈ dom card)
3028, 29syl 17 . . . 4 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → 𝑖 ∈ dom card)
31 carden2 9404 . . . 4 (((𝑎𝑆) ∈ dom card ∧ 𝑖 ∈ dom card) → ((card‘(𝑎𝑆)) = (card‘𝑖) ↔ (𝑎𝑆) ≈ 𝑖))
3226, 30, 31syl2anc 587 . . 3 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → ((card‘(𝑎𝑆)) = (card‘𝑖) ↔ (𝑎𝑆) ≈ 𝑖))
332adantrr 716 . . . . 5 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → ∃!𝑗𝑆 (𝑗𝑆) ≈ 𝑖)
34 ineq1 4134 . . . . . . 7 (𝑗 = 𝑎 → (𝑗𝑆) = (𝑎𝑆))
3534breq1d 5043 . . . . . 6 (𝑗 = 𝑎 → ((𝑗𝑆) ≈ 𝑖 ↔ (𝑎𝑆) ≈ 𝑖))
3635riota2 7122 . . . . 5 ((𝑎𝑆 ∧ ∃!𝑗𝑆 (𝑗𝑆) ≈ 𝑖) → ((𝑎𝑆) ≈ 𝑖 ↔ (𝑗𝑆 (𝑗𝑆) ≈ 𝑖) = 𝑎))
3719, 33, 36syl2anc 587 . . . 4 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → ((𝑎𝑆) ≈ 𝑖 ↔ (𝑗𝑆 (𝑗𝑆) ≈ 𝑖) = 𝑎))
38 eqcom 2808 . . . 4 ((𝑗𝑆 (𝑗𝑆) ≈ 𝑖) = 𝑎𝑎 = (𝑗𝑆 (𝑗𝑆) ≈ 𝑖))
3937, 38syl6bb 290 . . 3 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → ((𝑎𝑆) ≈ 𝑖𝑎 = (𝑗𝑆 (𝑗𝑆) ≈ 𝑖)))
4017, 32, 393bitrd 308 . 2 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → (𝑖 = (card‘(𝑎𝑆)) ↔ 𝑎 = (𝑗𝑆 (𝑗𝑆) ≈ 𝑖)))
411, 4, 11, 40f1o2d 7383 1 ((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) → 𝐶:ω–1-1-onto𝑆)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  ∃!wreu 3111  cin 3883  wss 3884   class class class wbr 5033  cmpt 5113  dom cdm 5523  Oncon0 6163  1-1-ontowf1o 6327  cfv 6328  crio 7096  ωcom 7564  cen 8493  Fincfn 8496  cardccrd 9352
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-om 7565  df-wrecs 7934  df-recs 7995  df-1o 8089  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-card 9356
This theorem is referenced by:  fin23lem27  9743  fin23lem28  9755  fin23lem30  9757  isf32lem6  9773  isf32lem7  9774  isf32lem8  9775
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