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Theorem ttukey2g 9221
 Description: The Teichmüller-Tukey Lemma ttukey 9223 with a slightly stronger conclusion: we can set up the maximal element of 𝐴 so that it also contains some given 𝐵 ∈ 𝐴 as a subset. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
ttukey2g (( 𝐴 ∈ dom card ∧ 𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥𝐴 (𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem ttukey2g
Dummy variables 𝑤 𝑓 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 difss 3699 . . . 4 ( 𝐴𝐵) ⊆ 𝐴
2 ssnum 8745 . . . 4 (( 𝐴 ∈ dom card ∧ ( 𝐴𝐵) ⊆ 𝐴) → ( 𝐴𝐵) ∈ dom card)
31, 2mpan2 703 . . 3 ( 𝐴 ∈ dom card → ( 𝐴𝐵) ∈ dom card)
4 isnum3 8663 . . . . 5 (( 𝐴𝐵) ∈ dom card ↔ (card‘( 𝐴𝐵)) ≈ ( 𝐴𝐵))
5 bren 7850 . . . . 5 ((card‘( 𝐴𝐵)) ≈ ( 𝐴𝐵) ↔ ∃𝑓 𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
64, 5bitri 263 . . . 4 (( 𝐴𝐵) ∈ dom card ↔ ∃𝑓 𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
7 simp1 1054 . . . . . . 7 ((𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) ∧ 𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → 𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
8 simp2 1055 . . . . . . 7 ((𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) ∧ 𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → 𝐵𝐴)
9 simp3 1056 . . . . . . 7 ((𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) ∧ 𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))
10 dmeq 5246 . . . . . . . . . . 11 (𝑤 = 𝑧 → dom 𝑤 = dom 𝑧)
1110unieqd 4382 . . . . . . . . . . 11 (𝑤 = 𝑧 dom 𝑤 = dom 𝑧)
1210, 11eqeq12d 2625 . . . . . . . . . 10 (𝑤 = 𝑧 → (dom 𝑤 = dom 𝑤 ↔ dom 𝑧 = dom 𝑧))
1310eqeq1d 2612 . . . . . . . . . . 11 (𝑤 = 𝑧 → (dom 𝑤 = ∅ ↔ dom 𝑧 = ∅))
14 rneq 5272 . . . . . . . . . . . 12 (𝑤 = 𝑧 → ran 𝑤 = ran 𝑧)
1514unieqd 4382 . . . . . . . . . . 11 (𝑤 = 𝑧 ran 𝑤 = ran 𝑧)
1613, 15ifbieq2d 4061 . . . . . . . . . 10 (𝑤 = 𝑧 → if(dom 𝑤 = ∅, 𝐵, ran 𝑤) = if(dom 𝑧 = ∅, 𝐵, ran 𝑧))
17 id 22 . . . . . . . . . . . 12 (𝑤 = 𝑧𝑤 = 𝑧)
1817, 11fveq12d 6109 . . . . . . . . . . 11 (𝑤 = 𝑧 → (𝑤 dom 𝑤) = (𝑧 dom 𝑧))
1911fveq2d 6107 . . . . . . . . . . . . . . 15 (𝑤 = 𝑧 → (𝑓 dom 𝑤) = (𝑓 dom 𝑧))
2019sneqd 4137 . . . . . . . . . . . . . 14 (𝑤 = 𝑧 → {(𝑓 dom 𝑤)} = {(𝑓 dom 𝑧)})
2118, 20uneq12d 3730 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → ((𝑤 dom 𝑤) ∪ {(𝑓 dom 𝑤)}) = ((𝑧 dom 𝑧) ∪ {(𝑓 dom 𝑧)}))
2221eleq1d 2672 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (((𝑤 dom 𝑤) ∪ {(𝑓 dom 𝑤)}) ∈ 𝐴 ↔ ((𝑧 dom 𝑧) ∪ {(𝑓 dom 𝑧)}) ∈ 𝐴))
2322, 20ifbieq1d 4059 . . . . . . . . . . 11 (𝑤 = 𝑧 → if(((𝑤 dom 𝑤) ∪ {(𝑓 dom 𝑤)}) ∈ 𝐴, {(𝑓 dom 𝑤)}, ∅) = if(((𝑧 dom 𝑧) ∪ {(𝑓 dom 𝑧)}) ∈ 𝐴, {(𝑓 dom 𝑧)}, ∅))
2418, 23uneq12d 3730 . . . . . . . . . 10 (𝑤 = 𝑧 → ((𝑤 dom 𝑤) ∪ if(((𝑤 dom 𝑤) ∪ {(𝑓 dom 𝑤)}) ∈ 𝐴, {(𝑓 dom 𝑤)}, ∅)) = ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝑓 dom 𝑧)}) ∈ 𝐴, {(𝑓 dom 𝑧)}, ∅)))
2512, 16, 24ifbieq12d 4063 . . . . . . . . 9 (𝑤 = 𝑧 → if(dom 𝑤 = dom 𝑤, if(dom 𝑤 = ∅, 𝐵, ran 𝑤), ((𝑤 dom 𝑤) ∪ if(((𝑤 dom 𝑤) ∪ {(𝑓 dom 𝑤)}) ∈ 𝐴, {(𝑓 dom 𝑤)}, ∅))) = if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝑓 dom 𝑧)}) ∈ 𝐴, {(𝑓 dom 𝑧)}, ∅))))
2625cbvmptv 4678 . . . . . . . 8 (𝑤 ∈ V ↦ if(dom 𝑤 = dom 𝑤, if(dom 𝑤 = ∅, 𝐵, ran 𝑤), ((𝑤 dom 𝑤) ∪ if(((𝑤 dom 𝑤) ∪ {(𝑓 dom 𝑤)}) ∈ 𝐴, {(𝑓 dom 𝑤)}, ∅)))) = (𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝑓 dom 𝑧)}) ∈ 𝐴, {(𝑓 dom 𝑧)}, ∅))))
27 recseq 7357 . . . . . . . 8 ((𝑤 ∈ V ↦ if(dom 𝑤 = dom 𝑤, if(dom 𝑤 = ∅, 𝐵, ran 𝑤), ((𝑤 dom 𝑤) ∪ if(((𝑤 dom 𝑤) ∪ {(𝑓 dom 𝑤)}) ∈ 𝐴, {(𝑓 dom 𝑤)}, ∅)))) = (𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝑓 dom 𝑧)}) ∈ 𝐴, {(𝑓 dom 𝑧)}, ∅)))) → recs((𝑤 ∈ V ↦ if(dom 𝑤 = dom 𝑤, if(dom 𝑤 = ∅, 𝐵, ran 𝑤), ((𝑤 dom 𝑤) ∪ if(((𝑤 dom 𝑤) ∪ {(𝑓 dom 𝑤)}) ∈ 𝐴, {(𝑓 dom 𝑤)}, ∅))))) = recs((𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝑓 dom 𝑧)}) ∈ 𝐴, {(𝑓 dom 𝑧)}, ∅))))))
2826, 27ax-mp 5 . . . . . . 7 recs((𝑤 ∈ V ↦ if(dom 𝑤 = dom 𝑤, if(dom 𝑤 = ∅, 𝐵, ran 𝑤), ((𝑤 dom 𝑤) ∪ if(((𝑤 dom 𝑤) ∪ {(𝑓 dom 𝑤)}) ∈ 𝐴, {(𝑓 dom 𝑤)}, ∅))))) = recs((𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝑓 dom 𝑧)}) ∈ 𝐴, {(𝑓 dom 𝑧)}, ∅)))))
297, 8, 9, 28ttukeylem7 9220 . . . . . 6 ((𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) ∧ 𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥𝐴 (𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦))
30293expib 1260 . . . . 5 (𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) → ((𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥𝐴 (𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦)))
3130exlimiv 1845 . . . 4 (∃𝑓 𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) → ((𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥𝐴 (𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦)))
326, 31sylbi 206 . . 3 (( 𝐴𝐵) ∈ dom card → ((𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥𝐴 (𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦)))
333, 32syl 17 . 2 ( 𝐴 ∈ dom card → ((𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥𝐴 (𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦)))
34333impib 1254 1 (( 𝐴 ∈ dom card ∧ 𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥𝐴 (𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540   ⊊ wpss 3541  ∅c0 3874  ifcif 4036  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372   class class class wbr 4583   ↦ cmpt 4643  dom cdm 5038  ran crn 5039  –1-1-onto→wf1o 5803  ‘cfv 5804  recscrecs 7354   ≈ 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-fin 7845  df-card 8648 This theorem is referenced by:  ttukeyg  9222
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