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Theorem ttukey2g 9927
Description: The Teichmüller-Tukey Lemma ttukey 9929 with a slightly stronger conclusion: we can set up the maximal element of 𝐴 so that it also contains some given 𝐵𝐴 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 4083 . . . 4 ( 𝐴𝐵) ⊆ 𝐴
2 ssnum 9454 . . . 4 (( 𝐴 ∈ dom card ∧ ( 𝐴𝐵) ⊆ 𝐴) → ( 𝐴𝐵) ∈ dom card)
31, 2mpan2 690 . . 3 ( 𝐴 ∈ dom card → ( 𝐴𝐵) ∈ dom card)
4 isnum3 9371 . . . . 5 (( 𝐴𝐵) ∈ dom card ↔ (card‘( 𝐴𝐵)) ≈ ( 𝐴𝐵))
5 bren 8505 . . . . 5 ((card‘( 𝐴𝐵)) ≈ ( 𝐴𝐵) ↔ ∃𝑓 𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
64, 5bitri 278 . . . 4 (( 𝐴𝐵) ∈ dom card ↔ ∃𝑓 𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
7 simp1 1133 . . . . . . 7 ((𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) ∧ 𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → 𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
8 simp2 1134 . . . . . . 7 ((𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) ∧ 𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → 𝐵𝐴)
9 simp3 1135 . . . . . . 7 ((𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) ∧ 𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))
10 dmeq 5749 . . . . . . . . . . 11 (𝑤 = 𝑧 → dom 𝑤 = dom 𝑧)
1110unieqd 4827 . . . . . . . . . . 11 (𝑤 = 𝑧 dom 𝑤 = dom 𝑧)
1210, 11eqeq12d 2838 . . . . . . . . . 10 (𝑤 = 𝑧 → (dom 𝑤 = dom 𝑤 ↔ dom 𝑧 = dom 𝑧))
1310eqeq1d 2824 . . . . . . . . . . 11 (𝑤 = 𝑧 → (dom 𝑤 = ∅ ↔ dom 𝑧 = ∅))
14 rneq 5783 . . . . . . . . . . . 12 (𝑤 = 𝑧 → ran 𝑤 = ran 𝑧)
1514unieqd 4827 . . . . . . . . . . 11 (𝑤 = 𝑧 ran 𝑤 = ran 𝑧)
1613, 15ifbieq2d 4464 . . . . . . . . . 10 (𝑤 = 𝑧 → if(dom 𝑤 = ∅, 𝐵, ran 𝑤) = if(dom 𝑧 = ∅, 𝐵, ran 𝑧))
17 id 22 . . . . . . . . . . . 12 (𝑤 = 𝑧𝑤 = 𝑧)
1817, 11fveq12d 6659 . . . . . . . . . . 11 (𝑤 = 𝑧 → (𝑤 dom 𝑤) = (𝑧 dom 𝑧))
1911fveq2d 6656 . . . . . . . . . . . . . . 15 (𝑤 = 𝑧 → (𝑓 dom 𝑤) = (𝑓 dom 𝑧))
2019sneqd 4551 . . . . . . . . . . . . . 14 (𝑤 = 𝑧 → {(𝑓 dom 𝑤)} = {(𝑓 dom 𝑧)})
2118, 20uneq12d 4115 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → ((𝑤 dom 𝑤) ∪ {(𝑓 dom 𝑤)}) = ((𝑧 dom 𝑧) ∪ {(𝑓 dom 𝑧)}))
2221eleq1d 2898 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (((𝑤 dom 𝑤) ∪ {(𝑓 dom 𝑤)}) ∈ 𝐴 ↔ ((𝑧 dom 𝑧) ∪ {(𝑓 dom 𝑧)}) ∈ 𝐴))
2322, 20ifbieq1d 4462 . . . . . . . . . . 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 4115 . . . . . . . . . 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 4466 . . . . . . . . 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 5145 . . . . . . . 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 7997 . . . . . . . 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 9926 . . . . . 6 ((𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) ∧ 𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥𝐴 (𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦))
30293expib 1119 . . . . 5 (𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) → ((𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥𝐴 (𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦)))
3130exlimiv 1931 . . . 4 (∃𝑓 𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) → ((𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥𝐴 (𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦)))
326, 31sylbi 220 . . 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 1113 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 209  wa 399  w3a 1084  wal 1536   = wceq 1538  wex 1781  wcel 2114  wral 3130  wrex 3131  Vcvv 3469  cdif 3905  cun 3906  cin 3907  wss 3908  wpss 3909  c0 4265  ifcif 4439  𝒫 cpw 4511  {csn 4539   cuni 4813   class class class wbr 5042  cmpt 5122  dom cdm 5532  ran crn 5533  1-1-ontowf1o 6333  cfv 6334  recscrecs 7994  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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
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 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-se 5492  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isom 6343  df-riota 7098  df-om 7566  df-wrecs 7934  df-recs 7995  df-1o 8089  df-er 8276  df-en 8497  df-dom 8498  df-fin 8500  df-card 9356
This theorem is referenced by:  ttukeyg  9928
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