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Theorem wfrlem14 7955
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 7954 . 2 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝐶 Fn (dom 𝐹 ∪ {𝑧}))
6 elun 4079 . . . 4 (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) ↔ (𝑦 ∈ dom 𝐹𝑦 ∈ {𝑧}))
7 velsn 4544 . . . . 5 (𝑦 ∈ {𝑧} ↔ 𝑦 = 𝑧)
87orbi2i 910 . . . 4 ((𝑦 ∈ dom 𝐹𝑦 ∈ {𝑧}) ↔ (𝑦 ∈ dom 𝐹𝑦 = 𝑧))
96, 8bitri 278 . . 3 (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) ↔ (𝑦 ∈ dom 𝐹𝑦 = 𝑧))
101, 2, 3wfrlem12 7953 . . . . . . 7 (𝑦 ∈ dom 𝐹 → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
11 fnfun 6427 . . . . . . . 8 (𝐶 Fn (dom 𝐹 ∪ {𝑧}) → Fun 𝐶)
12 ssun1 4102 . . . . . . . . . 10 𝐹 ⊆ (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
1312, 4sseqtrri 3955 . . . . . . . . 9 𝐹𝐶
14 funssfv 6670 . . . . . . . . . 10 ((Fun 𝐶𝐹𝐶𝑦 ∈ dom 𝐹) → (𝐶𝑦) = (𝐹𝑦))
153wfrdmcl 7950 . . . . . . . . . . . 12 (𝑦 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝐹)
16 fun2ssres 6373 . . . . . . . . . . . 12 ((Fun 𝐶𝐹𝐶 ∧ Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝐹) → (𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))
1715, 16syl3an3 1162 . . . . . . . . . . 11 ((Fun 𝐶𝐹𝐶𝑦 ∈ dom 𝐹) → (𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))
1817fveq2d 6653 . . . . . . . . . 10 ((Fun 𝐶𝐹𝐶𝑦 ∈ dom 𝐹) → (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
1914, 18eqeq12d 2817 . . . . . . . . 9 ((Fun 𝐶𝐹𝐶𝑦 ∈ dom 𝐹) → ((𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
2013, 19mp3an2 1446 . . . . . . . 8 ((Fun 𝐶𝑦 ∈ dom 𝐹) → ((𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
2111, 20sylan 583 . . . . . . 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 249 . . . . . 6 ((𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ 𝑦 ∈ dom 𝐹) → (𝑦 ∈ dom 𝐹 → (𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
2322ex 416 . . . . 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 4565 . . . . . . 7 𝑧 ∈ {𝑧}
26 elun2 4107 . . . . . . 7 (𝑧 ∈ {𝑧} → 𝑧 ∈ (dom 𝐹 ∪ {𝑧}))
2725, 26ax-mp 5 . . . . . 6 𝑧 ∈ (dom 𝐹 ∪ {𝑧})
284reseq1i 5818 . . . . . . . . . . . . 13 (𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)) = ((𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) ↾ Pred(𝑅, 𝐴, 𝑧))
29 resundir 5837 . . . . . . . . . . . . 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 5513 . . . . . . . . . . . . . . . . 17 (𝑅 We 𝐴𝑅 Fr 𝐴)
311, 30ax-mp 5 . . . . . . . . . . . . . . . 16 𝑅 Fr 𝐴
32 predfrirr 6149 . . . . . . . . . . . . . . . 16 (𝑅 Fr 𝐴 → ¬ 𝑧 ∈ Pred(𝑅, 𝐴, 𝑧))
33 ressnop0 6896 . . . . . . . . . . . . . . . 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 4090 . . . . . . . . . . . . . 14 ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) ∪ ({⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ↾ Pred(𝑅, 𝐴, 𝑧))) = ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) ∪ ∅)
36 un0 4301 . . . . . . . . . . . . . 14 ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) ∪ ∅) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))
3735, 36eqtri 2824 . . . . . . . . . . . . 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 2828 . . . . . . . . . . . 12 (𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))
3938fveq2i 6652 . . . . . . . . . . 11 (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))
4039opeq2i 4772 . . . . . . . . . 10 𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ = ⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩
41 opex 5324 . . . . . . . . . . 11 𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ ∈ V
4241elsn 4543 . . . . . . . . . 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 234 . . . . . . . . 9 𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ ∈ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}
44 elun2 4107 . . . . . . . . 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 2892 . . . . . . 7 𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ ∈ 𝐶
47 fnopfvb 6698 . . . . . . 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 261 . . . . . 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 6649 . . . . . 6 (𝑦 = 𝑧 → (𝐶𝑦) = (𝐶𝑧))
51 predeq3 6124 . . . . . . . 8 (𝑦 = 𝑧 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑧))
5251reseq2d 5822 . . . . . . 7 (𝑦 = 𝑧 → (𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))
5352fveq2d 6653 . . . . . 6 (𝑦 = 𝑧 → (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧))))
5450, 53eqeq12d 2817 . . . . 5 (𝑦 = 𝑧 → ((𝐶𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐶𝑧) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))))
5549, 54syl5ibrcom 250 . . . 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, 56syl5bi 245 . 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 209  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2112  cdif 3881  cun 3882  wss 3884  c0 4246  {csn 4528  cop 4534   Fr wfr 5479   Se wse 5480   We wwe 5481  dom cdm 5523  cres 5525  Predcpred 6119  Fun wfun 6322   Fn wfn 6323  cfv 6328  wrecscwrecs 7933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336  df-wrecs 7934
This theorem is referenced by:  wfrlem15  7956
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