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Theorem ttukey2g 10351
Description: The Teichmüller-Tukey Lemma ttukey 10353 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 4076 . . . 4 ( 𝐴𝐵) ⊆ 𝐴
2 ssnum 9874 . . . 4 (( 𝐴 ∈ dom card ∧ ( 𝐴𝐵) ⊆ 𝐴) → ( 𝐴𝐵) ∈ dom card)
31, 2mpan2 688 . . 3 ( 𝐴 ∈ dom card → ( 𝐴𝐵) ∈ dom card)
4 isnum3 9789 . . . . 5 (( 𝐴𝐵) ∈ dom card ↔ (card‘( 𝐴𝐵)) ≈ ( 𝐴𝐵))
5 bren 8792 . . . . 5 ((card‘( 𝐴𝐵)) ≈ ( 𝐴𝐵) ↔ ∃𝑓 𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
64, 5bitri 274 . . . 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 5832 . . . . . . . . . . 11 (𝑤 = 𝑧 → dom 𝑤 = dom 𝑧)
1110unieqd 4863 . . . . . . . . . . 11 (𝑤 = 𝑧 dom 𝑤 = dom 𝑧)
1210, 11eqeq12d 2752 . . . . . . . . . 10 (𝑤 = 𝑧 → (dom 𝑤 = dom 𝑤 ↔ dom 𝑧 = dom 𝑧))
1310eqeq1d 2738 . . . . . . . . . . 11 (𝑤 = 𝑧 → (dom 𝑤 = ∅ ↔ dom 𝑧 = ∅))
14 rneq 5864 . . . . . . . . . . . 12 (𝑤 = 𝑧 → ran 𝑤 = ran 𝑧)
1514unieqd 4863 . . . . . . . . . . 11 (𝑤 = 𝑧 ran 𝑤 = ran 𝑧)
1613, 15ifbieq2d 4496 . . . . . . . . . 10 (𝑤 = 𝑧 → if(dom 𝑤 = ∅, 𝐵, ran 𝑤) = if(dom 𝑧 = ∅, 𝐵, ran 𝑧))
17 id 22 . . . . . . . . . . . 12 (𝑤 = 𝑧𝑤 = 𝑧)
1817, 11fveq12d 6818 . . . . . . . . . . 11 (𝑤 = 𝑧 → (𝑤 dom 𝑤) = (𝑧 dom 𝑧))
1911fveq2d 6815 . . . . . . . . . . . . . . 15 (𝑤 = 𝑧 → (𝑓 dom 𝑤) = (𝑓 dom 𝑧))
2019sneqd 4582 . . . . . . . . . . . . . 14 (𝑤 = 𝑧 → {(𝑓 dom 𝑤)} = {(𝑓 dom 𝑧)})
2118, 20uneq12d 4108 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → ((𝑤 dom 𝑤) ∪ {(𝑓 dom 𝑤)}) = ((𝑧 dom 𝑧) ∪ {(𝑓 dom 𝑧)}))
2221eleq1d 2821 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (((𝑤 dom 𝑤) ∪ {(𝑓 dom 𝑤)}) ∈ 𝐴 ↔ ((𝑧 dom 𝑧) ∪ {(𝑓 dom 𝑧)}) ∈ 𝐴))
2322, 20ifbieq1d 4494 . . . . . . . . . . 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 4108 . . . . . . . . . 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 4498 . . . . . . . . 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 5199 . . . . . . . 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 8253 . . . . . . . 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 10350 . . . . . 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 1932 . . . 4 (∃𝑓 𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) → ((𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥𝐴 (𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦)))
326, 31sylbi 216 . . 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 205  wa 396  w3a 1086  wal 1538   = wceq 1540  wex 1780  wcel 2105  wral 3061  wrex 3070  Vcvv 3440  cdif 3893  cun 3894  cin 3895  wss 3896  wpss 3897  c0 4266  ifcif 4470  𝒫 cpw 4544  {csn 4570   cuni 4849   class class class wbr 5086  cmpt 5169  dom cdm 5607  ran crn 5608  1-1-ontowf1o 6464  cfv 6465  recscrecs 8249  cen 8779  Fincfn 8782  cardccrd 9770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5223  ax-sep 5237  ax-nul 5244  ax-pow 5302  ax-pr 5366  ax-un 7629
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3442  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-pss 3915  df-nul 4267  df-if 4471  df-pw 4546  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4850  df-int 4892  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5170  df-tr 5204  df-id 5506  df-eprel 5512  df-po 5520  df-so 5521  df-fr 5562  df-se 5563  df-we 5564  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-pred 6224  df-ord 6291  df-on 6292  df-lim 6293  df-suc 6294  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fo 6471  df-f1o 6472  df-fv 6473  df-isom 6474  df-riota 7273  df-ov 7319  df-om 7759  df-2nd 7878  df-frecs 8145  df-wrecs 8176  df-recs 8250  df-1o 8345  df-er 8547  df-en 8783  df-dom 8784  df-fin 8786  df-card 9774
This theorem is referenced by:  ttukeyg  10352
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