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Theorem ttukeylem4 9926
Description: Lemma for ttukey 9932. (Contributed by Mario Carneiro, 15-May-2015.)
Hypotheses
Ref Expression
ttukeylem.1 (𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
ttukeylem.2 (𝜑𝐵𝐴)
ttukeylem.3 (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))
ttukeylem.4 𝐺 = 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)}, ∅)))))
Assertion
Ref Expression
ttukeylem4 (𝜑 → (𝐺‘∅) = 𝐵)
Distinct variable groups:   𝑥,𝑧,𝐺   𝜑,𝑧   𝑥,𝐴,𝑧   𝑥,𝐵,𝑧   𝑥,𝐹,𝑧
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ttukeylem4
StepHypRef Expression
1 0elon 6241 . . 3 ∅ ∈ On
2 ttukeylem.1 . . . 4 (𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
3 ttukeylem.2 . . . 4 (𝜑𝐵𝐴)
4 ttukeylem.3 . . . 4 (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))
5 ttukeylem.4 . . . 4 𝐺 = 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)}, ∅)))))
62, 3, 4, 5ttukeylem3 9925 . . 3 ((𝜑 ∧ ∅ ∈ On) → (𝐺‘∅) = if(∅ = ∅, if(∅ = ∅, 𝐵, (𝐺 “ ∅)), ((𝐺 ∅) ∪ if(((𝐺 ∅) ∪ {(𝐹 ∅)}) ∈ 𝐴, {(𝐹 ∅)}, ∅))))
71, 6mpan2 687 . 2 (𝜑 → (𝐺‘∅) = if(∅ = ∅, if(∅ = ∅, 𝐵, (𝐺 “ ∅)), ((𝐺 ∅) ∪ if(((𝐺 ∅) ∪ {(𝐹 ∅)}) ∈ 𝐴, {(𝐹 ∅)}, ∅))))
8 uni0 4863 . . . . 5 ∅ = ∅
98eqcomi 2833 . . . 4 ∅ =
109iftruei 4476 . . 3 if(∅ = ∅, if(∅ = ∅, 𝐵, (𝐺 “ ∅)), ((𝐺 ∅) ∪ if(((𝐺 ∅) ∪ {(𝐹 ∅)}) ∈ 𝐴, {(𝐹 ∅)}, ∅))) = if(∅ = ∅, 𝐵, (𝐺 “ ∅))
11 eqid 2824 . . . 4 ∅ = ∅
1211iftruei 4476 . . 3 if(∅ = ∅, 𝐵, (𝐺 “ ∅)) = 𝐵
1310, 12eqtri 2848 . 2 if(∅ = ∅, if(∅ = ∅, 𝐵, (𝐺 “ ∅)), ((𝐺 ∅) ∪ if(((𝐺 ∅) ∪ {(𝐹 ∅)}) ∈ 𝐴, {(𝐹 ∅)}, ∅))) = 𝐵
147, 13syl6eq 2876 1 (𝜑 → (𝐺‘∅) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wal 1528   = wceq 1530  wcel 2106  Vcvv 3499  cdif 3936  cun 3937  cin 3938  wss 3939  c0 4294  ifcif 4469  𝒫 cpw 4541  {csn 4563   cuni 4836  cmpt 5142  dom cdm 5553  ran crn 5554  cima 5556  Oncon0 6188  1-1-ontowf1o 6350  cfv 6351  recscrecs 8001  Fincfn 8501  cardccrd 9356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-wrecs 7941  df-recs 8002
This theorem is referenced by:  ttukeylem7  9929
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