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Theorem frrlem1 8310
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 Paul Chapman, 21-Apr-2012.)
Hypothesis
Ref Expression
frrlem1.1 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
Assertion
Ref Expression
frrlem1 𝐵 = {𝑔 ∣ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝑤𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))}
Distinct variable groups:   𝐴,𝑓,𝑔,𝑤,𝑥,𝑦,𝑧   𝑓,𝐺,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 frrlem1
StepHypRef Expression
1 frrlem1.1 . 2 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
2 fneq1 6660 . . . . . 6 (𝑓 = 𝑔 → (𝑓 Fn 𝑥𝑔 Fn 𝑥))
3 fveq1 6906 . . . . . . . 8 (𝑓 = 𝑔 → (𝑓𝑦) = (𝑔𝑦))
4 reseq1 5994 . . . . . . . . 9 (𝑓 = 𝑔 → (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))
54oveq2d 7447 . . . . . . . 8 (𝑓 = 𝑔 → (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑦𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))))
63, 5eqeq12d 2751 . . . . . . 7 (𝑓 = 𝑔 → ((𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝑔𝑦) = (𝑦𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))))
76ralbidv 3176 . . . . . 6 (𝑓 = 𝑔 → (∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑦𝑥 (𝑔𝑦) = (𝑦𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))))
82, 73anbi13d 1437 . . . . 5 (𝑓 = 𝑔 → ((𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ (𝑔 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝑦𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))))))
98exbidv 1919 . . . 4 (𝑓 = 𝑔 → (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ∃𝑥(𝑔 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝑦𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))))))
10 fneq2 6661 . . . . . 6 (𝑥 = 𝑧 → (𝑔 Fn 𝑥𝑔 Fn 𝑧))
11 sseq1 4021 . . . . . . 7 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
12 sseq2 4022 . . . . . . . . 9 (𝑥 = 𝑧 → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑧))
1312raleqbi1dv 3336 . . . . . . . 8 (𝑥 = 𝑧 → (∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ ∀𝑦𝑧 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑧))
14 predeq3 6327 . . . . . . . . . 10 (𝑦 = 𝑤 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑤))
1514sseq1d 4027 . . . . . . . . 9 (𝑦 = 𝑤 → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑧 ↔ Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧))
1615cbvralvw 3235 . . . . . . . 8 (∀𝑦𝑧 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑧 ↔ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)
1713, 16bitrdi 287 . . . . . . 7 (𝑥 = 𝑧 → (∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧))
1811, 17anbi12d 632 . . . . . 6 (𝑥 = 𝑧 → ((𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ↔ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))
19 raleq 3321 . . . . . . 7 (𝑥 = 𝑧 → (∀𝑦𝑥 (𝑔𝑦) = (𝑦𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑦𝑧 (𝑔𝑦) = (𝑦𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))))
20 fveq2 6907 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑔𝑦) = (𝑔𝑤))
21 id 22 . . . . . . . . . 10 (𝑦 = 𝑤𝑦 = 𝑤)
2214reseq2d 6000 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))
2321, 22oveq12d 7449 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑦𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑤𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))
2420, 23eqeq12d 2751 . . . . . . . 8 (𝑦 = 𝑤 → ((𝑔𝑦) = (𝑦𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝑔𝑤) = (𝑤𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))))
2524cbvralvw 3235 . . . . . . 7 (∀𝑦𝑧 (𝑔𝑦) = (𝑦𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑤𝑧 (𝑔𝑤) = (𝑤𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))
2619, 25bitrdi 287 . . . . . 6 (𝑥 = 𝑧 → (∀𝑦𝑥 (𝑔𝑦) = (𝑦𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑤𝑧 (𝑔𝑤) = (𝑤𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))))
2710, 18, 263anbi123d 1435 . . . . 5 (𝑥 = 𝑧 → ((𝑔 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝑦𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ (𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝑤𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))))
2827cbvexvw 2034 . . . 4 (∃𝑥(𝑔 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑔𝑦) = (𝑦𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝑤𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))))
299, 28bitrdi 287 . . 3 (𝑓 = 𝑔 → (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝑤𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))))
3029cbvabv 2810 . 2 {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} = {𝑔 ∣ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝑤𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))}
311, 30eqtri 2763 1 𝐵 = {𝑔 ∣ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝑤𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))}
Colors of variables: wff setvar class
Syntax hints:  wa 395  w3a 1086   = wceq 1537  wex 1776  {cab 2712  wral 3059  wss 3963  cres 5691  Predcpred 6322   Fn wfn 6558  cfv 6563  (class class class)co 7431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-ov 7434
This theorem is referenced by:  frrlem2  8311  frrlem3  8312  frrlem4  8313  frrlem8  8317  frrlem12  8321  frrlem13  8322  fpr1  8327  fprresex  8334  frr1  9797  frr2  9798
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