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Theorem wfrlem1 7652
Description: Lemma for well-founded recursion. The final item we are interested in is the union of acceptable functions 𝐵. This lemma just changes bound variables for later use. (Contributed by Scott Fenton, 21-Apr-2011.)
Hypothesis
Ref Expression
wfrlem1.1 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
Assertion
Ref Expression
wfrlem1 𝐵 = {𝑔 ∣ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))}
Distinct variable groups:   𝐴,𝑓,𝑔,𝑤,𝑥,𝑦,𝑧   𝑓,𝐹,𝑔,𝑤,𝑥,𝑦,𝑧   𝑅,𝑓,𝑔,𝑤,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐵(𝑥,𝑦,𝑧,𝑤,𝑓,𝑔)

Proof of Theorem wfrlem1
StepHypRef Expression
1 wfrlem1.1 . 2 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
2 fneq1 6193 . . . . . 6 (𝑓 = 𝑔 → (𝑓 Fn 𝑥𝑔 Fn 𝑥))
3 fveq1 6410 . . . . . . . 8 (𝑓 = 𝑔 → (𝑓𝑦) = (𝑔𝑦))
4 reseq1 5598 . . . . . . . . 9 (𝑓 = 𝑔 → (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))
54fveq2d 6415 . . . . . . . 8 (𝑓 = 𝑔 → (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))))
63, 5eqeq12d 2828 . . . . . . 7 (𝑓 = 𝑔 → ((𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝑔𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))))
76ralbidv 3181 . . . . . 6 (𝑓 = 𝑔 → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑦𝑥 (𝑔𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))))
82, 73anbi13d 1555 . . . . 5 (𝑓 = 𝑔 → ((𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ (𝑔 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))))))
98exbidv 2012 . . . 4 (𝑓 = 𝑔 → (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ∃𝑥(𝑔 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))))))
10 fneq2 6194 . . . . . 6 (𝑥 = 𝑧 → (𝑔 Fn 𝑥𝑔 Fn 𝑧))
11 sseq1 3830 . . . . . . 7 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
12 sseq2 3831 . . . . . . . . 9 (𝑥 = 𝑧 → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑧))
1312raleqbi1dv 3342 . . . . . . . 8 (𝑥 = 𝑧 → (∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ ∀𝑦𝑧 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑧))
14 predeq3 5904 . . . . . . . . . 10 (𝑦 = 𝑤 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑤))
1514sseq1d 3836 . . . . . . . . 9 (𝑦 = 𝑤 → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑧 ↔ Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧))
1615cbvralv 3367 . . . . . . . 8 (∀𝑦𝑧 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑧 ↔ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)
1713, 16syl6bb 278 . . . . . . 7 (𝑥 = 𝑧 → (∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧))
1811, 17anbi12d 618 . . . . . 6 (𝑥 = 𝑧 → ((𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ↔ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))
19 raleq 3334 . . . . . . 7 (𝑥 = 𝑧 → (∀𝑦𝑥 (𝑔𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑦𝑧 (𝑔𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))))
20 fveq2 6411 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑔𝑦) = (𝑔𝑤))
2114reseq2d 5604 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))
2221fveq2d 6415 . . . . . . . . 9 (𝑦 = 𝑤 → (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))
2320, 22eqeq12d 2828 . . . . . . . 8 (𝑦 = 𝑤 → ((𝑔𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝑔𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))))
2423cbvralv 3367 . . . . . . 7 (∀𝑦𝑧 (𝑔𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))
2519, 24syl6bb 278 . . . . . 6 (𝑥 = 𝑧 → (∀𝑦𝑥 (𝑔𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))))
2610, 18, 253anbi123d 1553 . . . . 5 (𝑥 = 𝑧 → ((𝑔 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ (𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))))
2726cbvexvw 2138 . . . 4 (∃𝑥(𝑔 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))))
289, 27syl6bb 278 . . 3 (𝑓 = 𝑔 → (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))))
2928cbvabv 2938 . 2 {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} = {𝑔 ∣ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))}
301, 29eqtri 2835 1 𝐵 = {𝑔 ∣ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))}
Colors of variables: wff setvar class
Syntax hints:  wa 384  w3a 1100   = wceq 1637  wex 1859  {cab 2799  wral 3103  wss 3776  cres 5320  Predcpred 5899   Fn wfn 6099  cfv 6104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3400  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-sn 4378  df-pr 4380  df-op 4384  df-uni 4638  df-br 4852  df-opab 4914  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-pred 5900  df-iota 6067  df-fun 6106  df-fn 6107  df-fv 6112
This theorem is referenced by:  wfrlem2  7653  wfrlem3  7654  wfrlem3a  7655  wfrlem4  7656  wfrlem4OLD  7657  wfrdmcl  7662
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