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Theorem ttukey2g 10554
Description: The Teichmüller-Tukey Lemma ttukey 10556 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 4146 . . . 4 ( 𝐴𝐵) ⊆ 𝐴
2 ssnum 10077 . . . 4 (( 𝐴 ∈ dom card ∧ ( 𝐴𝐵) ⊆ 𝐴) → ( 𝐴𝐵) ∈ dom card)
31, 2mpan2 691 . . 3 ( 𝐴 ∈ dom card → ( 𝐴𝐵) ∈ dom card)
4 isnum3 9992 . . . . 5 (( 𝐴𝐵) ∈ dom card ↔ (card‘( 𝐴𝐵)) ≈ ( 𝐴𝐵))
5 bren 8994 . . . . 5 ((card‘( 𝐴𝐵)) ≈ ( 𝐴𝐵) ↔ ∃𝑓 𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
64, 5bitri 275 . . . 4 (( 𝐴𝐵) ∈ dom card ↔ ∃𝑓 𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
7 simp1 1135 . . . . . . 7 ((𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) ∧ 𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → 𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
8 simp2 1136 . . . . . . 7 ((𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) ∧ 𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → 𝐵𝐴)
9 simp3 1137 . . . . . . 7 ((𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) ∧ 𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))
10 dmeq 5917 . . . . . . . . . . 11 (𝑤 = 𝑧 → dom 𝑤 = dom 𝑧)
1110unieqd 4925 . . . . . . . . . . 11 (𝑤 = 𝑧 dom 𝑤 = dom 𝑧)
1210, 11eqeq12d 2751 . . . . . . . . . 10 (𝑤 = 𝑧 → (dom 𝑤 = dom 𝑤 ↔ dom 𝑧 = dom 𝑧))
1310eqeq1d 2737 . . . . . . . . . . 11 (𝑤 = 𝑧 → (dom 𝑤 = ∅ ↔ dom 𝑧 = ∅))
14 rneq 5950 . . . . . . . . . . . 12 (𝑤 = 𝑧 → ran 𝑤 = ran 𝑧)
1514unieqd 4925 . . . . . . . . . . 11 (𝑤 = 𝑧 ran 𝑤 = ran 𝑧)
1613, 15ifbieq2d 4557 . . . . . . . . . 10 (𝑤 = 𝑧 → if(dom 𝑤 = ∅, 𝐵, ran 𝑤) = if(dom 𝑧 = ∅, 𝐵, ran 𝑧))
17 id 22 . . . . . . . . . . . 12 (𝑤 = 𝑧𝑤 = 𝑧)
1817, 11fveq12d 6914 . . . . . . . . . . 11 (𝑤 = 𝑧 → (𝑤 dom 𝑤) = (𝑧 dom 𝑧))
1911fveq2d 6911 . . . . . . . . . . . . . . 15 (𝑤 = 𝑧 → (𝑓 dom 𝑤) = (𝑓 dom 𝑧))
2019sneqd 4643 . . . . . . . . . . . . . 14 (𝑤 = 𝑧 → {(𝑓 dom 𝑤)} = {(𝑓 dom 𝑧)})
2118, 20uneq12d 4179 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → ((𝑤 dom 𝑤) ∪ {(𝑓 dom 𝑤)}) = ((𝑧 dom 𝑧) ∪ {(𝑓 dom 𝑧)}))
2221eleq1d 2824 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (((𝑤 dom 𝑤) ∪ {(𝑓 dom 𝑤)}) ∈ 𝐴 ↔ ((𝑧 dom 𝑧) ∪ {(𝑓 dom 𝑧)}) ∈ 𝐴))
2322, 20ifbieq1d 4555 . . . . . . . . . . 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 4179 . . . . . . . . . 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 4559 . . . . . . . . 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 5261 . . . . . . . 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 8413 . . . . . . . 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 10553 . . . . . 6 ((𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) ∧ 𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥𝐴 (𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦))
30293expib 1121 . . . . 5 (𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) → ((𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥𝐴 (𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦)))
3130exlimiv 1928 . . . 4 (∃𝑓 𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) → ((𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥𝐴 (𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦)))
326, 31sylbi 217 . . 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 1115 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 206  wa 395  w3a 1086  wal 1535   = wceq 1537  wex 1776  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  cdif 3960  cun 3961  cin 3962  wss 3963  wpss 3964  c0 4339  ifcif 4531  𝒫 cpw 4605  {csn 4631   cuni 4912   class class class wbr 5148  cmpt 5231  dom cdm 5689  ran crn 5690  1-1-ontowf1o 6562  cfv 6563  recscrecs 8409  cen 8981  Fincfn 8984  cardccrd 9973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-fin 8988  df-card 9977
This theorem is referenced by:  ttukeyg  10555
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