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Theorem fin23lem22 9545
Description: Lemma for fin23 9607 but could be used elsewhere if we find a good name for it. Explicit construction of a bijection (actually an isomorphism, see fin23lem27 9546) 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 9544 . . 3 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∃!𝑗𝑆 (𝑗𝑆) ≈ 𝑖)
3 riotacl 6949 . . 3 (∃!𝑗𝑆 (𝑗𝑆) ≈ 𝑖 → (𝑗𝑆 (𝑗𝑆) ≈ 𝑖) ∈ 𝑆)
42, 3syl 17 . 2 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → (𝑗𝑆 (𝑗𝑆) ≈ 𝑖) ∈ 𝑆)
5 simpll 755 . . . 4 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑎𝑆) → 𝑆 ⊆ ω)
6 simpr 477 . . . 4 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑎𝑆) → 𝑎𝑆)
75, 6sseldd 3852 . . 3 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑎𝑆) → 𝑎 ∈ ω)
8 nnfi 8504 . . 3 (𝑎 ∈ ω → 𝑎 ∈ Fin)
9 infi 8535 . . 3 (𝑎 ∈ Fin → (𝑎𝑆) ∈ Fin)
10 ficardom 9182 . . 3 ((𝑎𝑆) ∈ Fin → (card‘(𝑎𝑆)) ∈ ω)
117, 8, 9, 104syl 19 . 2 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑎𝑆) → (card‘(𝑎𝑆)) ∈ ω)
12 cardnn 9184 . . . . . . 7 (𝑖 ∈ ω → (card‘𝑖) = 𝑖)
1312eqcomd 2777 . . . . . 6 (𝑖 ∈ ω → 𝑖 = (card‘𝑖))
1413eqeq1d 2773 . . . . 5 (𝑖 ∈ ω → (𝑖 = (card‘(𝑎𝑆)) ↔ (card‘𝑖) = (card‘(𝑎𝑆))))
15 eqcom 2778 . . . . 5 ((card‘𝑖) = (card‘(𝑎𝑆)) ↔ (card‘(𝑎𝑆)) = (card‘𝑖))
1614, 15syl6bb 279 . . . 4 (𝑖 ∈ ω → (𝑖 = (card‘(𝑎𝑆)) ↔ (card‘(𝑎𝑆)) = (card‘𝑖)))
1716ad2antrl 716 . . 3 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → (𝑖 = (card‘(𝑎𝑆)) ↔ (card‘(𝑎𝑆)) = (card‘𝑖)))
18 simpll 755 . . . . . . 7 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → 𝑆 ⊆ ω)
19 simprr 761 . . . . . . 7 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → 𝑎𝑆)
2018, 19sseldd 3852 . . . . . 6 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → 𝑎 ∈ ω)
21 nnon 7400 . . . . . 6 (𝑎 ∈ ω → 𝑎 ∈ On)
22 onenon 9170 . . . . . 6 (𝑎 ∈ On → 𝑎 ∈ dom card)
2320, 21, 223syl 18 . . . . 5 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → 𝑎 ∈ dom card)
24 inss1 4086 . . . . 5 (𝑎𝑆) ⊆ 𝑎
25 ssnum 9257 . . . . 5 ((𝑎 ∈ dom card ∧ (𝑎𝑆) ⊆ 𝑎) → (𝑎𝑆) ∈ dom card)
2623, 24, 25sylancl 578 . . . 4 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → (𝑎𝑆) ∈ dom card)
27 nnon 7400 . . . . . 6 (𝑖 ∈ ω → 𝑖 ∈ On)
2827ad2antrl 716 . . . . 5 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → 𝑖 ∈ On)
29 onenon 9170 . . . . 5 (𝑖 ∈ On → 𝑖 ∈ dom card)
3028, 29syl 17 . . . 4 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → 𝑖 ∈ dom card)
31 carden2 9208 . . . 4 (((𝑎𝑆) ∈ dom card ∧ 𝑖 ∈ dom card) → ((card‘(𝑎𝑆)) = (card‘𝑖) ↔ (𝑎𝑆) ≈ 𝑖))
3226, 30, 31syl2anc 576 . . 3 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → ((card‘(𝑎𝑆)) = (card‘𝑖) ↔ (𝑎𝑆) ≈ 𝑖))
332adantrr 705 . . . . 5 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → ∃!𝑗𝑆 (𝑗𝑆) ≈ 𝑖)
34 ineq1 4062 . . . . . . 7 (𝑗 = 𝑎 → (𝑗𝑆) = (𝑎𝑆))
3534breq1d 4935 . . . . . 6 (𝑗 = 𝑎 → ((𝑗𝑆) ≈ 𝑖 ↔ (𝑎𝑆) ≈ 𝑖))
3635riota2 6957 . . . . 5 ((𝑎𝑆 ∧ ∃!𝑗𝑆 (𝑗𝑆) ≈ 𝑖) → ((𝑎𝑆) ≈ 𝑖 ↔ (𝑗𝑆 (𝑗𝑆) ≈ 𝑖) = 𝑎))
3719, 33, 36syl2anc 576 . . . 4 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → ((𝑎𝑆) ≈ 𝑖 ↔ (𝑗𝑆 (𝑗𝑆) ≈ 𝑖) = 𝑎))
38 eqcom 2778 . . . 4 ((𝑗𝑆 (𝑗𝑆) ≈ 𝑖) = 𝑎𝑎 = (𝑗𝑆 (𝑗𝑆) ≈ 𝑖))
3937, 38syl6bb 279 . . 3 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → ((𝑎𝑆) ≈ 𝑖𝑎 = (𝑗𝑆 (𝑗𝑆) ≈ 𝑖)))
4017, 32, 393bitrd 297 . 2 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → (𝑖 = (card‘(𝑎𝑆)) ↔ 𝑎 = (𝑗𝑆 (𝑗𝑆) ≈ 𝑖)))
411, 4, 11, 40f1o2d 7215 1 ((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) → 𝐶:ω–1-1-onto𝑆)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387   = wceq 1508  wcel 2051  ∃!wreu 3083  cin 3821  wss 3822   class class class wbr 4925  cmpt 5004  dom cdm 5403  Oncon0 6026  1-1-ontowf1o 6184  cfv 6185  crio 6934  ωcom 7394  cen 8301  Fincfn 8304  cardccrd 9156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-reu 3088  df-rmo 3089  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-pss 3838  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-int 4746  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-se 5363  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-isom 6194  df-riota 6935  df-om 7395  df-wrecs 7748  df-recs 7810  df-1o 7903  df-er 8087  df-en 8305  df-dom 8306  df-sdom 8307  df-fin 8308  df-card 9160
This theorem is referenced by:  fin23lem27  9546  fin23lem28  9558  fin23lem30  9560  isf32lem6  9576  isf32lem7  9577  isf32lem8  9578
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