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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 ↦ (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 9031 . . 3 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → ∃!𝑗𝑆 (𝑗𝑆) ≈ 𝑖)
3 riotacl 6525 . . 3 (∃!𝑗𝑆 (𝑗𝑆) ≈ 𝑖 → (𝑗𝑆 (𝑗𝑆) ≈ 𝑖) ∈ 𝑆)
42, 3syl 17 . 2 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑖 ∈ ω) → (𝑗𝑆 (𝑗𝑆) ≈ 𝑖) ∈ 𝑆)
5 simpll 786 . . . 4 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑎𝑆) → 𝑆 ⊆ ω)
6 simpr 476 . . . 4 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑎𝑆) → 𝑎𝑆)
75, 6sseldd 3569 . . 3 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑎𝑆) → 𝑎 ∈ ω)
8 nnfi 8038 . . 3 (𝑎 ∈ ω → 𝑎 ∈ Fin)
9 infi 8069 . . 3 (𝑎 ∈ Fin → (𝑎𝑆) ∈ Fin)
10 ficardom 8670 . . 3 ((𝑎𝑆) ∈ Fin → (card‘(𝑎𝑆)) ∈ ω)
117, 8, 9, 104syl 19 . 2 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ 𝑎𝑆) → (card‘(𝑎𝑆)) ∈ ω)
12 cardnn 8672 . . . . . . 7 (𝑖 ∈ ω → (card‘𝑖) = 𝑖)
1312eqcomd 2616 . . . . . 6 (𝑖 ∈ ω → 𝑖 = (card‘𝑖))
1413eqeq1d 2612 . . . . 5 (𝑖 ∈ ω → (𝑖 = (card‘(𝑎𝑆)) ↔ (card‘𝑖) = (card‘(𝑎𝑆))))
15 eqcom 2617 . . . . 5 ((card‘𝑖) = (card‘(𝑎𝑆)) ↔ (card‘(𝑎𝑆)) = (card‘𝑖))
1614, 15syl6bb 275 . . . 4 (𝑖 ∈ ω → (𝑖 = (card‘(𝑎𝑆)) ↔ (card‘(𝑎𝑆)) = (card‘𝑖)))
1716ad2antrl 760 . . 3 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → (𝑖 = (card‘(𝑎𝑆)) ↔ (card‘(𝑎𝑆)) = (card‘𝑖)))
18 simpll 786 . . . . . . 7 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → 𝑆 ⊆ ω)
19 simprr 792 . . . . . . 7 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → 𝑎𝑆)
2018, 19sseldd 3569 . . . . . 6 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → 𝑎 ∈ ω)
21 nnon 6963 . . . . . 6 (𝑎 ∈ ω → 𝑎 ∈ On)
22 onenon 8658 . . . . . 6 (𝑎 ∈ On → 𝑎 ∈ dom card)
2320, 21, 223syl 18 . . . . 5 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → 𝑎 ∈ dom card)
24 inss1 3795 . . . . 5 (𝑎𝑆) ⊆ 𝑎
25 ssnum 8745 . . . . 5 ((𝑎 ∈ dom card ∧ (𝑎𝑆) ⊆ 𝑎) → (𝑎𝑆) ∈ dom card)
2623, 24, 25sylancl 693 . . . 4 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → (𝑎𝑆) ∈ dom card)
27 nnon 6963 . . . . . 6 (𝑖 ∈ ω → 𝑖 ∈ On)
2827ad2antrl 760 . . . . 5 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → 𝑖 ∈ On)
29 onenon 8658 . . . . 5 (𝑖 ∈ On → 𝑖 ∈ dom card)
3028, 29syl 17 . . . 4 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → 𝑖 ∈ dom card)
31 carden2 8696 . . . 4 (((𝑎𝑆) ∈ dom card ∧ 𝑖 ∈ dom card) → ((card‘(𝑎𝑆)) = (card‘𝑖) ↔ (𝑎𝑆) ≈ 𝑖))
3226, 30, 31syl2anc 691 . . 3 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → ((card‘(𝑎𝑆)) = (card‘𝑖) ↔ (𝑎𝑆) ≈ 𝑖))
332adantrr 749 . . . . 5 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → ∃!𝑗𝑆 (𝑗𝑆) ≈ 𝑖)
34 ineq1 3769 . . . . . . 7 (𝑗 = 𝑎 → (𝑗𝑆) = (𝑎𝑆))
3534breq1d 4593 . . . . . 6 (𝑗 = 𝑎 → ((𝑗𝑆) ≈ 𝑖 ↔ (𝑎𝑆) ≈ 𝑖))
3635riota2 6533 . . . . 5 ((𝑎𝑆 ∧ ∃!𝑗𝑆 (𝑗𝑆) ≈ 𝑖) → ((𝑎𝑆) ≈ 𝑖 ↔ (𝑗𝑆 (𝑗𝑆) ≈ 𝑖) = 𝑎))
3719, 33, 36syl2anc 691 . . . 4 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → ((𝑎𝑆) ≈ 𝑖 ↔ (𝑗𝑆 (𝑗𝑆) ≈ 𝑖) = 𝑎))
38 eqcom 2617 . . . 4 ((𝑗𝑆 (𝑗𝑆) ≈ 𝑖) = 𝑎𝑎 = (𝑗𝑆 (𝑗𝑆) ≈ 𝑖))
3937, 38syl6bb 275 . . 3 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → ((𝑎𝑆) ≈ 𝑖𝑎 = (𝑗𝑆 (𝑗𝑆) ≈ 𝑖)))
4017, 32, 393bitrd 293 . 2 (((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) ∧ (𝑖 ∈ ω ∧ 𝑎𝑆)) → (𝑖 = (card‘(𝑎𝑆)) ↔ 𝑎 = (𝑗𝑆 (𝑗𝑆) ≈ 𝑖)))
411, 4, 11, 40f1o2d 6785 1 ((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) → 𝐶:ω–1-1-onto𝑆)
 Colors of variables: wff setvar class 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
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