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Theorem ttukey2g 9530
Description: The Teichmüller-Tukey Lemma ttukey 9532 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 3880 . . . 4 ( 𝐴𝐵) ⊆ 𝐴
2 ssnum 9052 . . . 4 (( 𝐴 ∈ dom card ∧ ( 𝐴𝐵) ⊆ 𝐴) → ( 𝐴𝐵) ∈ dom card)
31, 2mpan2 709 . . 3 ( 𝐴 ∈ dom card → ( 𝐴𝐵) ∈ dom card)
4 isnum3 8970 . . . . 5 (( 𝐴𝐵) ∈ dom card ↔ (card‘( 𝐴𝐵)) ≈ ( 𝐴𝐵))
5 bren 8130 . . . . 5 ((card‘( 𝐴𝐵)) ≈ ( 𝐴𝐵) ↔ ∃𝑓 𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
64, 5bitri 264 . . . 4 (( 𝐴𝐵) ∈ dom card ↔ ∃𝑓 𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
7 simp1 1131 . . . . . . 7 ((𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) ∧ 𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → 𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
8 simp2 1132 . . . . . . 7 ((𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) ∧ 𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → 𝐵𝐴)
9 simp3 1133 . . . . . . 7 ((𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) ∧ 𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))
10 dmeq 5479 . . . . . . . . . . 11 (𝑤 = 𝑧 → dom 𝑤 = dom 𝑧)
1110unieqd 4598 . . . . . . . . . . 11 (𝑤 = 𝑧 dom 𝑤 = dom 𝑧)
1210, 11eqeq12d 2775 . . . . . . . . . 10 (𝑤 = 𝑧 → (dom 𝑤 = dom 𝑤 ↔ dom 𝑧 = dom 𝑧))
1310eqeq1d 2762 . . . . . . . . . . 11 (𝑤 = 𝑧 → (dom 𝑤 = ∅ ↔ dom 𝑧 = ∅))
14 rneq 5506 . . . . . . . . . . . 12 (𝑤 = 𝑧 → ran 𝑤 = ran 𝑧)
1514unieqd 4598 . . . . . . . . . . 11 (𝑤 = 𝑧 ran 𝑤 = ran 𝑧)
1613, 15ifbieq2d 4255 . . . . . . . . . 10 (𝑤 = 𝑧 → if(dom 𝑤 = ∅, 𝐵, ran 𝑤) = if(dom 𝑧 = ∅, 𝐵, ran 𝑧))
17 id 22 . . . . . . . . . . . 12 (𝑤 = 𝑧𝑤 = 𝑧)
1817, 11fveq12d 6358 . . . . . . . . . . 11 (𝑤 = 𝑧 → (𝑤 dom 𝑤) = (𝑧 dom 𝑧))
1911fveq2d 6356 . . . . . . . . . . . . . . 15 (𝑤 = 𝑧 → (𝑓 dom 𝑤) = (𝑓 dom 𝑧))
2019sneqd 4333 . . . . . . . . . . . . . 14 (𝑤 = 𝑧 → {(𝑓 dom 𝑤)} = {(𝑓 dom 𝑧)})
2118, 20uneq12d 3911 . . . . . . . . . . . . 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 4253 . . . . . . . . . . 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 3911 . . . . . . . . . 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 4257 . . . . . . . . 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 4902 . . . . . . . 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 7639 . . . . . . . 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 9529 . . . . . 6 ((𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) ∧ 𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥𝐴 (𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦))
30293expib 1117 . . . . 5 (𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) → ((𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥𝐴 (𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦)))
3130exlimiv 2007 . . . 4 (∃𝑓 𝑓:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) → ((𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥𝐴 (𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦)))
326, 31sylbi 207 . . 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 1109 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 196  wa 383  w3a 1072  wal 1630   = wceq 1632  wex 1853  wcel 2139  wral 3050  wrex 3051  Vcvv 3340  cdif 3712  cun 3713  cin 3714  wss 3715  wpss 3716  c0 4058  ifcif 4230  𝒫 cpw 4302  {csn 4321   cuni 4588   class class class wbr 4804  cmpt 4881  dom cdm 5266  ran crn 5267  1-1-ontowf1o 6048  cfv 6049  recscrecs 7636  cen 8118  Fincfn 8121  cardccrd 8951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-om 7231  df-wrecs 7576  df-recs 7637  df-1o 7729  df-er 7911  df-en 8122  df-dom 8123  df-fin 8125  df-card 8955
This theorem is referenced by:  ttukeyg  9531
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