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Theorem wfrlem14OLD 8223
Description: Lemma for well-ordered recursion. Compute the value of 𝐶. Obsolete as of 18-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 21-Apr-2011.)
Hypotheses
Ref Expression
wfrlem13OLD.1 𝑅 We 𝐴
wfrlem13OLD.2 𝑅 Se 𝐴
wfrlem13OLD.3 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
wfrlem13OLD.4 𝐶 = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
Assertion
Ref Expression
wfrlem14OLD (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) → (𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
Distinct variable groups:   𝑦,𝐴,𝑧   𝑦,𝐹,𝑧   𝑦,𝐺   𝑦,𝑅,𝑧   𝑦,𝐶
Allowed substitution hints:   𝐶(𝑧)   𝐺(𝑧)

Proof of Theorem wfrlem14OLD
StepHypRef Expression
1 wfrlem13OLD.1 . . 3 𝑅 We 𝐴
2 wfrlem13OLD.2 . . 3 𝑅 Se 𝐴
3 wfrlem13OLD.3 . . 3 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
4 wfrlem13OLD.4 . . 3 𝐶 = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
51, 2, 3, 4wfrlem13OLD 8222 . 2 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝐶 Fn (dom 𝐹 ∪ {𝑧}))
6 elun 4095 . . . 4 (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) ↔ (𝑦 ∈ dom 𝐹𝑦 ∈ {𝑧}))
7 velsn 4589 . . . . 5 (𝑦 ∈ {𝑧} ↔ 𝑦 = 𝑧)
87orbi2i 910 . . . 4 ((𝑦 ∈ dom 𝐹𝑦 ∈ {𝑧}) ↔ (𝑦 ∈ dom 𝐹𝑦 = 𝑧))
96, 8bitri 274 . . 3 (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) ↔ (𝑦 ∈ dom 𝐹𝑦 = 𝑧))
101, 2, 3wfrlem12OLD 8221 . . . . . . 7 (𝑦 ∈ dom 𝐹 → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
11 fnfun 6585 . . . . . . . 8 (𝐶 Fn (dom 𝐹 ∪ {𝑧}) → Fun 𝐶)
12 ssun1 4119 . . . . . . . . . 10 𝐹 ⊆ (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
1312, 4sseqtrri 3969 . . . . . . . . 9 𝐹𝐶
14 funssfv 6846 . . . . . . . . . 10 ((Fun 𝐶𝐹𝐶𝑦 ∈ dom 𝐹) → (𝐶𝑦) = (𝐹𝑦))
153wfrdmclOLD 8218 . . . . . . . . . . . 12 (𝑦 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝐹)
16 fun2ssres 6529 . . . . . . . . . . . 12 ((Fun 𝐶𝐹𝐶 ∧ Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝐹) → (𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))
1715, 16syl3an3 1164 . . . . . . . . . . 11 ((Fun 𝐶𝐹𝐶𝑦 ∈ dom 𝐹) → (𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))
1817fveq2d 6829 . . . . . . . . . 10 ((Fun 𝐶𝐹𝐶𝑦 ∈ dom 𝐹) → (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
1914, 18eqeq12d 2752 . . . . . . . . 9 ((Fun 𝐶𝐹𝐶𝑦 ∈ dom 𝐹) → ((𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
2013, 19mp3an2 1448 . . . . . . . 8 ((Fun 𝐶𝑦 ∈ dom 𝐹) → ((𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
2111, 20sylan 580 . . . . . . 7 ((𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ 𝑦 ∈ dom 𝐹) → ((𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
2210, 21syl5ibr 245 . . . . . 6 ((𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ 𝑦 ∈ dom 𝐹) → (𝑦 ∈ dom 𝐹 → (𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
2322ex 413 . . . . 5 (𝐶 Fn (dom 𝐹 ∪ {𝑧}) → (𝑦 ∈ dom 𝐹 → (𝑦 ∈ dom 𝐹 → (𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))))))
2423pm2.43d 53 . . . 4 (𝐶 Fn (dom 𝐹 ∪ {𝑧}) → (𝑦 ∈ dom 𝐹 → (𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
25 vsnid 4610 . . . . . . 7 𝑧 ∈ {𝑧}
26 elun2 4124 . . . . . . 7 (𝑧 ∈ {𝑧} → 𝑧 ∈ (dom 𝐹 ∪ {𝑧}))
2725, 26ax-mp 5 . . . . . 6 𝑧 ∈ (dom 𝐹 ∪ {𝑧})
284reseq1i 5919 . . . . . . . . . . . . 13 (𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)) = ((𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) ↾ Pred(𝑅, 𝐴, 𝑧))
29 resundir 5938 . . . . . . . . . . . . 13 ((𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) ↾ Pred(𝑅, 𝐴, 𝑧)) = ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) ∪ ({⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ↾ Pred(𝑅, 𝐴, 𝑧)))
30 wefr 5610 . . . . . . . . . . . . . . . . 17 (𝑅 We 𝐴𝑅 Fr 𝐴)
311, 30ax-mp 5 . . . . . . . . . . . . . . . 16 𝑅 Fr 𝐴
32 predfrirr 6273 . . . . . . . . . . . . . . . 16 (𝑅 Fr 𝐴 → ¬ 𝑧 ∈ Pred(𝑅, 𝐴, 𝑧))
33 ressnop0 7081 . . . . . . . . . . . . . . . 16 𝑧 ∈ Pred(𝑅, 𝐴, 𝑧) → ({⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ↾ Pred(𝑅, 𝐴, 𝑧)) = ∅)
3431, 32, 33mp2b 10 . . . . . . . . . . . . . . 15 ({⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ↾ Pred(𝑅, 𝐴, 𝑧)) = ∅
3534uneq2i 4107 . . . . . . . . . . . . . 14 ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) ∪ ({⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ↾ Pred(𝑅, 𝐴, 𝑧))) = ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) ∪ ∅)
36 un0 4337 . . . . . . . . . . . . . 14 ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) ∪ ∅) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))
3735, 36eqtri 2764 . . . . . . . . . . . . 13 ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) ∪ ({⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))
3828, 29, 373eqtri 2768 . . . . . . . . . . . 12 (𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))
3938fveq2i 6828 . . . . . . . . . . 11 (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))
4039opeq2i 4821 . . . . . . . . . 10 𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ = ⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩
41 opex 5409 . . . . . . . . . . 11 𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ ∈ V
4241elsn 4588 . . . . . . . . . 10 (⟨𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ ∈ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ↔ ⟨𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ = ⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩)
4340, 42mpbir 230 . . . . . . . . 9 𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ ∈ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}
44 elun2 4124 . . . . . . . . 9 (⟨𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ ∈ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} → ⟨𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ ∈ (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}))
4543, 44ax-mp 5 . . . . . . . 8 𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ ∈ (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
4645, 4eleqtrri 2836 . . . . . . 7 𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ ∈ 𝐶
47 fnopfvb 6879 . . . . . . 7 ((𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ 𝑧 ∈ (dom 𝐹 ∪ {𝑧})) → ((𝐶𝑧) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧))) ↔ ⟨𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ ∈ 𝐶))
4846, 47mpbiri 257 . . . . . 6 ((𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ 𝑧 ∈ (dom 𝐹 ∪ {𝑧})) → (𝐶𝑧) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧))))
4927, 48mpan2 688 . . . . 5 (𝐶 Fn (dom 𝐹 ∪ {𝑧}) → (𝐶𝑧) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧))))
50 fveq2 6825 . . . . . 6 (𝑦 = 𝑧 → (𝐶𝑦) = (𝐶𝑧))
51 predeq3 6242 . . . . . . . 8 (𝑦 = 𝑧 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑧))
5251reseq2d 5923 . . . . . . 7 (𝑦 = 𝑧 → (𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))
5352fveq2d 6829 . . . . . 6 (𝑦 = 𝑧 → (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧))))
5450, 53eqeq12d 2752 . . . . 5 (𝑦 = 𝑧 → ((𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐶𝑧) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))))
5549, 54syl5ibrcom 246 . . . 4 (𝐶 Fn (dom 𝐹 ∪ {𝑧}) → (𝑦 = 𝑧 → (𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
5624, 55jaod 856 . . 3 (𝐶 Fn (dom 𝐹 ∪ {𝑧}) → ((𝑦 ∈ dom 𝐹𝑦 = 𝑧) → (𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
579, 56biimtrid 241 . 2 (𝐶 Fn (dom 𝐹 ∪ {𝑧}) → (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) → (𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
585, 57syl 17 1 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) → (𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 844  w3a 1086   = wceq 1540  wcel 2105  cdif 3895  cun 3896  wss 3898  c0 4269  {csn 4573  cop 4579   Fr wfr 5572   Se wse 5573   We wwe 5574  dom cdm 5620  cres 5622  Predcpred 6237  Fun wfun 6473   Fn wfn 6474  cfv 6479  wrecscwrecs 8197
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-sep 5243  ax-nul 5250  ax-pr 5372  ax-un 7650
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  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-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-po 5532  df-so 5533  df-fr 5575  df-se 5576  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-fo 6485  df-fv 6487  df-ov 7340  df-2nd 7900  df-frecs 8167  df-wrecs 8198
This theorem is referenced by:  wfrlem15OLD  8224
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