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Theorem wfrlem14 7376
Description: Lemma for well-founded recursion. Compute the value of 𝐶. (Contributed by Scott Fenton, 21-Apr-2011.)
Hypotheses
Ref Expression
wfrlem13.1 𝑅 We 𝐴
wfrlem13.2 𝑅 Se 𝐴
wfrlem13.3 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
wfrlem13.4 𝐶 = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
Assertion
Ref Expression
wfrlem14 (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 wfrlem14
StepHypRef Expression
1 wfrlem13.1 . . 3 𝑅 We 𝐴
2 wfrlem13.2 . . 3 𝑅 Se 𝐴
3 wfrlem13.3 . . 3 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
4 wfrlem13.4 . . 3 𝐶 = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
51, 2, 3, 4wfrlem13 7375 . 2 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝐶 Fn (dom 𝐹 ∪ {𝑧}))
6 elun 3733 . . . 4 (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) ↔ (𝑦 ∈ dom 𝐹𝑦 ∈ {𝑧}))
7 velsn 4166 . . . . 5 (𝑦 ∈ {𝑧} ↔ 𝑦 = 𝑧)
87orbi2i 541 . . . 4 ((𝑦 ∈ dom 𝐹𝑦 ∈ {𝑧}) ↔ (𝑦 ∈ dom 𝐹𝑦 = 𝑧))
96, 8bitri 264 . . 3 (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) ↔ (𝑦 ∈ dom 𝐹𝑦 = 𝑧))
101, 2, 3wfrlem12 7374 . . . . . . 7 (𝑦 ∈ dom 𝐹 → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
11 fnfun 5948 . . . . . . . 8 (𝐶 Fn (dom 𝐹 ∪ {𝑧}) → Fun 𝐶)
12 ssun1 3756 . . . . . . . . . 10 𝐹 ⊆ (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
1312, 4sseqtr4i 3619 . . . . . . . . 9 𝐹𝐶
14 funssfv 6168 . . . . . . . . . 10 ((Fun 𝐶𝐹𝐶𝑦 ∈ dom 𝐹) → (𝐶𝑦) = (𝐹𝑦))
153wfrdmcl 7371 . . . . . . . . . . . 12 (𝑦 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝐹)
16 fun2ssres 5891 . . . . . . . . . . . 12 ((Fun 𝐶𝐹𝐶 ∧ Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝐹) → (𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))
1715, 16syl3an3 1358 . . . . . . . . . . 11 ((Fun 𝐶𝐹𝐶𝑦 ∈ dom 𝐹) → (𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))
1817fveq2d 6154 . . . . . . . . . 10 ((Fun 𝐶𝐹𝐶𝑦 ∈ dom 𝐹) → (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
1914, 18eqeq12d 2636 . . . . . . . . 9 ((Fun 𝐶𝐹𝐶𝑦 ∈ dom 𝐹) → ((𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
2013, 19mp3an2 1409 . . . . . . . 8 ((Fun 𝐶𝑦 ∈ dom 𝐹) → ((𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
2111, 20sylan 488 . . . . . . 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 236 . . . . . 6 ((𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ 𝑦 ∈ dom 𝐹) → (𝑦 ∈ dom 𝐹 → (𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
2322ex 450 . . . . 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 4182 . . . . . . 7 𝑧 ∈ {𝑧}
26 elun2 3761 . . . . . . 7 (𝑧 ∈ {𝑧} → 𝑧 ∈ (dom 𝐹 ∪ {𝑧}))
2725, 26ax-mp 5 . . . . . 6 𝑧 ∈ (dom 𝐹 ∪ {𝑧})
284reseq1i 5354 . . . . . . . . . . . . 13 (𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)) = ((𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) ↾ Pred(𝑅, 𝐴, 𝑧))
29 resundir 5372 . . . . . . . . . . . . 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 5066 . . . . . . . . . . . . . . . . 17 (𝑅 We 𝐴𝑅 Fr 𝐴)
311, 30ax-mp 5 . . . . . . . . . . . . . . . 16 𝑅 Fr 𝐴
32 predfrirr 5670 . . . . . . . . . . . . . . . 16 (𝑅 Fr 𝐴 → ¬ 𝑧 ∈ Pred(𝑅, 𝐴, 𝑧))
33 ressnop0 6377 . . . . . . . . . . . . . . . 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 3744 . . . . . . . . . . . . . 14 ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) ∪ ({⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ↾ Pred(𝑅, 𝐴, 𝑧))) = ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) ∪ ∅)
36 un0 3941 . . . . . . . . . . . . . 14 ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) ∪ ∅) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))
3735, 36eqtri 2643 . . . . . . . . . . . . 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 2647 . . . . . . . . . . . 12 (𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))
3938fveq2i 6153 . . . . . . . . . . 11 (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))
4039opeq2i 4376 . . . . . . . . . 10 𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ = ⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩
41 opex 4895 . . . . . . . . . . 11 𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ ∈ V
4241elsn 4165 . . . . . . . . . 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 221 . . . . . . . . 9 𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ ∈ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}
44 elun2 3761 . . . . . . . . 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 2697 . . . . . . 7 𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ ∈ 𝐶
47 fnopfvb 6196 . . . . . . 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 248 . . . . . 6 ((𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ 𝑧 ∈ (dom 𝐹 ∪ {𝑧})) → (𝐶𝑧) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧))))
4927, 48mpan2 706 . . . . 5 (𝐶 Fn (dom 𝐹 ∪ {𝑧}) → (𝐶𝑧) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧))))
50 fveq2 6150 . . . . . 6 (𝑦 = 𝑧 → (𝐶𝑦) = (𝐶𝑧))
51 predeq3 5645 . . . . . . . 8 (𝑦 = 𝑧 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑧))
5251reseq2d 5358 . . . . . . 7 (𝑦 = 𝑧 → (𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))
5352fveq2d 6154 . . . . . 6 (𝑦 = 𝑧 → (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧))))
5450, 53eqeq12d 2636 . . . . 5 (𝑦 = 𝑧 → ((𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐶𝑧) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))))
5549, 54syl5ibrcom 237 . . . 4 (𝐶 Fn (dom 𝐹 ∪ {𝑧}) → (𝑦 = 𝑧 → (𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
5624, 55jaod 395 . . 3 (𝐶 Fn (dom 𝐹 ∪ {𝑧}) → ((𝑦 ∈ dom 𝐹𝑦 = 𝑧) → (𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
579, 56syl5bi 232 . 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 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  cdif 3553  cun 3554  wss 3556  c0 3893  {csn 4150  cop 4156   Fr wfr 5032   Se wse 5033   We wwe 5034  dom cdm 5076  cres 5078  Predcpred 5640  Fun wfun 5843   Fn wfn 5844  cfv 5849  wrecscwrecs 7354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-se 5036  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-iota 5812  df-fun 5851  df-fn 5852  df-fv 5857  df-wrecs 7355
This theorem is referenced by:  wfrlem15  7377
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