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Theorem frrlem1 32095
Description: Lemma for 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 6184 . . . . . 6 (𝑓 = 𝑔 → (𝑓 Fn 𝑥𝑔 Fn 𝑥))
3 fveq1 6401 . . . . . . . 8 (𝑓 = 𝑔 → (𝑓𝑦) = (𝑔𝑦))
4 reseq1 5585 . . . . . . . . 9 (𝑓 = 𝑔 → (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))
54oveq2d 6884 . . . . . . . 8 (𝑓 = 𝑔 → (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑦𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))))
63, 5eqeq12d 2817 . . . . . . 7 (𝑓 = 𝑔 → ((𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝑔𝑦) = (𝑦𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))))
76ralbidv 3170 . . . . . 6 (𝑓 = 𝑔 → (∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑦𝑥 (𝑔𝑦) = (𝑦𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))))
82, 73anbi13d 1555 . . . . 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 2012 . . . 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 6185 . . . . . 6 (𝑥 = 𝑧 → (𝑔 Fn 𝑥𝑔 Fn 𝑧))
11 sseq1 3817 . . . . . . 7 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
12 sseq2 3818 . . . . . . . . 9 (𝑥 = 𝑧 → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑧))
1312raleqbi1dv 3331 . . . . . . . 8 (𝑥 = 𝑧 → (∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ ∀𝑦𝑧 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑧))
14 predeq3 5891 . . . . . . . . . 10 (𝑦 = 𝑤 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑤))
1514sseq1d 3823 . . . . . . . . 9 (𝑦 = 𝑤 → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑧 ↔ Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧))
1615cbvralv 3356 . . . . . . . 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 3323 . . . . . . 7 (𝑥 = 𝑧 → (∀𝑦𝑥 (𝑔𝑦) = (𝑦𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑦𝑧 (𝑔𝑦) = (𝑦𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))))
20 fveq2 6402 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑔𝑦) = (𝑔𝑤))
21 id 22 . . . . . . . . . 10 (𝑦 = 𝑤𝑦 = 𝑤)
2214reseq2d 5591 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))
2321, 22oveq12d 6886 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑦𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑤𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))
2420, 23eqeq12d 2817 . . . . . . . 8 (𝑦 = 𝑤 → ((𝑔𝑦) = (𝑦𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝑔𝑤) = (𝑤𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))))
2524cbvralv 3356 . . . . . . 7 (∀𝑦𝑧 (𝑔𝑦) = (𝑦𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑤𝑧 (𝑔𝑤) = (𝑤𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))
2619, 25syl6bb 278 . . . . . 6 (𝑥 = 𝑧 → (∀𝑦𝑥 (𝑔𝑦) = (𝑦𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑤𝑧 (𝑔𝑤) = (𝑤𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))))
2710, 18, 263anbi123d 1553 . . . . 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 2136 . . . 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, 28syl6bb 278 . . 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 2927 . 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 2824 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 384  w3a 1100   = wceq 1637  wex 1859  {cab 2788  wral 3092  wss 3763  cres 5307  Predcpred 5886   Fn wfn 6090  cfv 6095  (class class class)co 6868
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 2067  ax-7 2103  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781
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 2060  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ral 3097  df-rex 3098  df-rab 3101  df-v 3389  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-nul 4111  df-if 4274  df-sn 4365  df-pr 4367  df-op 4371  df-uni 4624  df-br 4838  df-opab 4900  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-pred 5887  df-iota 6058  df-fun 6097  df-fn 6098  df-fv 6103  df-ov 6871
This theorem is referenced by:  frrlem2  32096  frrlem3  32097  frrlem4  32098  frrlem5e  32103
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