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Theorem wfrlem14OLD 8322
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 8321 . 2 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝐶 Fn (dom 𝐹 ∪ {𝑧}))
6 elun 4149 . . . 4 (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) ↔ (𝑦 ∈ dom 𝐹𝑦 ∈ {𝑧}))
7 velsn 4645 . . . . 5 (𝑦 ∈ {𝑧} ↔ 𝑦 = 𝑧)
87orbi2i 912 . . . 4 ((𝑦 ∈ dom 𝐹𝑦 ∈ {𝑧}) ↔ (𝑦 ∈ dom 𝐹𝑦 = 𝑧))
96, 8bitri 275 . . 3 (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) ↔ (𝑦 ∈ dom 𝐹𝑦 = 𝑧))
101, 2, 3wfrlem12OLD 8320 . . . . . . 7 (𝑦 ∈ dom 𝐹 → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
11 fnfun 6650 . . . . . . . 8 (𝐶 Fn (dom 𝐹 ∪ {𝑧}) → Fun 𝐶)
12 ssun1 4173 . . . . . . . . . 10 𝐹 ⊆ (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
1312, 4sseqtrri 4020 . . . . . . . . 9 𝐹𝐶
14 funssfv 6913 . . . . . . . . . 10 ((Fun 𝐶𝐹𝐶𝑦 ∈ dom 𝐹) → (𝐶𝑦) = (𝐹𝑦))
153wfrdmclOLD 8317 . . . . . . . . . . . 12 (𝑦 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝐹)
16 fun2ssres 6594 . . . . . . . . . . . 12 ((Fun 𝐶𝐹𝐶 ∧ Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝐹) → (𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))
1715, 16syl3an3 1166 . . . . . . . . . . 11 ((Fun 𝐶𝐹𝐶𝑦 ∈ dom 𝐹) → (𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))
1817fveq2d 6896 . . . . . . . . . 10 ((Fun 𝐶𝐹𝐶𝑦 ∈ dom 𝐹) → (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
1914, 18eqeq12d 2749 . . . . . . . . 9 ((Fun 𝐶𝐹𝐶𝑦 ∈ dom 𝐹) → ((𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
2013, 19mp3an2 1450 . . . . . . . 8 ((Fun 𝐶𝑦 ∈ dom 𝐹) → ((𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
2111, 20sylan 581 . . . . . . 7 ((𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ 𝑦 ∈ dom 𝐹) → ((𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
2210, 21imbitrrid 245 . . . . . 6 ((𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ 𝑦 ∈ dom 𝐹) → (𝑦 ∈ dom 𝐹 → (𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
2322ex 414 . . . . 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 4666 . . . . . . 7 𝑧 ∈ {𝑧}
26 elun2 4178 . . . . . . 7 (𝑧 ∈ {𝑧} → 𝑧 ∈ (dom 𝐹 ∪ {𝑧}))
2725, 26ax-mp 5 . . . . . 6 𝑧 ∈ (dom 𝐹 ∪ {𝑧})
284reseq1i 5978 . . . . . . . . . . . . 13 (𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)) = ((𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) ↾ Pred(𝑅, 𝐴, 𝑧))
29 resundir 5997 . . . . . . . . . . . . 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 5667 . . . . . . . . . . . . . . . . 17 (𝑅 We 𝐴𝑅 Fr 𝐴)
311, 30ax-mp 5 . . . . . . . . . . . . . . . 16 𝑅 Fr 𝐴
32 predfrirr 6336 . . . . . . . . . . . . . . . 16 (𝑅 Fr 𝐴 → ¬ 𝑧 ∈ Pred(𝑅, 𝐴, 𝑧))
33 ressnop0 7151 . . . . . . . . . . . . . . . 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 4161 . . . . . . . . . . . . . 14 ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) ∪ ({⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ↾ Pred(𝑅, 𝐴, 𝑧))) = ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) ∪ ∅)
36 un0 4391 . . . . . . . . . . . . . 14 ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) ∪ ∅) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))
3735, 36eqtri 2761 . . . . . . . . . . . . 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 2765 . . . . . . . . . . . 12 (𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))
3938fveq2i 6895 . . . . . . . . . . 11 (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))
4039opeq2i 4878 . . . . . . . . . 10 𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ = ⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩
41 opex 5465 . . . . . . . . . . 11 𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ ∈ V
4241elsn 4644 . . . . . . . . . 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 4178 . . . . . . . . 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 2833 . . . . . . 7 𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ ∈ 𝐶
47 fnopfvb 6946 . . . . . . 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 258 . . . . . 6 ((𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ 𝑧 ∈ (dom 𝐹 ∪ {𝑧})) → (𝐶𝑧) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧))))
4927, 48mpan2 690 . . . . 5 (𝐶 Fn (dom 𝐹 ∪ {𝑧}) → (𝐶𝑧) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧))))
50 fveq2 6892 . . . . . 6 (𝑦 = 𝑧 → (𝐶𝑦) = (𝐶𝑧))
51 predeq3 6305 . . . . . . . 8 (𝑦 = 𝑧 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑧))
5251reseq2d 5982 . . . . . . 7 (𝑦 = 𝑧 → (𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))
5352fveq2d 6896 . . . . . 6 (𝑦 = 𝑧 → (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧))))
5450, 53eqeq12d 2749 . . . . 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 858 . . 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 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  cdif 3946  cun 3947  wss 3949  c0 4323  {csn 4629  cop 4635   Fr wfr 5629   Se wse 5630   We wwe 5631  dom cdm 5677  cres 5679  Predcpred 6300  Fun wfun 6538   Fn wfn 6539  cfv 6544  wrecscwrecs 8296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-ov 7412  df-2nd 7976  df-frecs 8266  df-wrecs 8297
This theorem is referenced by:  wfrlem15OLD  8323
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