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Theorem frecsuclem1 6017
 Description: Lemma for frecsuc 6021. (Contributed by Jim Kingdon, 13-Aug-2019.)
Hypothesis
Ref Expression
frecsuclem1.h 𝐺 = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})
Assertion
Ref Expression
frecsuclem1 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉𝐵 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝐵) = (𝐺‘(recs(𝐺) ↾ suc 𝐵)))
Distinct variable groups:   𝐴,𝑔,𝑚,𝑥,𝑧   𝐵,𝑔,𝑚,𝑥,𝑧   𝑔,𝐹,𝑚,𝑥,𝑧   𝑔,𝐺,𝑚,𝑥,𝑧   𝑔,𝑉,𝑚,𝑥
Allowed substitution hint:   𝑉(𝑧)

Proof of Theorem frecsuclem1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-frec 6008 . . . . . 6 frec(𝐹, 𝐴) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})) ↾ ω)
2 frecsuclem1.h . . . . . . . 8 𝐺 = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})
3 recseq 5951 . . . . . . . 8 (𝐺 = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}) → recs(𝐺) = recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})))
42, 3ax-mp 7 . . . . . . 7 recs(𝐺) = recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}))
54reseq1i 4635 . . . . . 6 (recs(𝐺) ↾ ω) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})) ↾ ω)
61, 5eqtr4i 2079 . . . . 5 frec(𝐹, 𝐴) = (recs(𝐺) ↾ ω)
76fveq1i 5206 . . . 4 (frec(𝐹, 𝐴)‘suc 𝐵) = ((recs(𝐺) ↾ ω)‘suc 𝐵)
8 peano2 4345 . . . . 5 (𝐵 ∈ ω → suc 𝐵 ∈ ω)
9 fvres 5225 . . . . 5 (suc 𝐵 ∈ ω → ((recs(𝐺) ↾ ω)‘suc 𝐵) = (recs(𝐺)‘suc 𝐵))
108, 9syl 14 . . . 4 (𝐵 ∈ ω → ((recs(𝐺) ↾ ω)‘suc 𝐵) = (recs(𝐺)‘suc 𝐵))
117, 10syl5eq 2100 . . 3 (𝐵 ∈ ω → (frec(𝐹, 𝐴)‘suc 𝐵) = (recs(𝐺)‘suc 𝐵))
12113ad2ant3 938 . 2 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉𝐵 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝐵) = (recs(𝐺)‘suc 𝐵))
13 nnon 4359 . . . . 5 (suc 𝐵 ∈ ω → suc 𝐵 ∈ On)
148, 13syl 14 . . . 4 (𝐵 ∈ ω → suc 𝐵 ∈ On)
15 eqid 2056 . . . . 5 recs(𝐺) = recs(𝐺)
162frectfr 6015 . . . . 5 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → ∀𝑦(Fun 𝐺 ∧ (𝐺𝑦) ∈ V))
1715, 16tfri2d 5980 . . . 4 (((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) ∧ suc 𝐵 ∈ On) → (recs(𝐺)‘suc 𝐵) = (𝐺‘(recs(𝐺) ↾ suc 𝐵)))
1814, 17sylan2 274 . . 3 (((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) ∧ 𝐵 ∈ ω) → (recs(𝐺)‘suc 𝐵) = (𝐺‘(recs(𝐺) ↾ suc 𝐵)))
19183impa 1110 . 2 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉𝐵 ∈ ω) → (recs(𝐺)‘suc 𝐵) = (𝐺‘(recs(𝐺) ↾ suc 𝐵)))
2012, 19eqtrd 2088 1 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉𝐵 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝐵) = (𝐺‘(recs(𝐺) ↾ suc 𝐵)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ∨ wo 639   ∧ w3a 896  ∀wal 1257   = wceq 1259   ∈ wcel 1409  {cab 2042  ∃wrex 2324  Vcvv 2574  ∅c0 3251   ↦ cmpt 3845  Oncon0 4127  suc csuc 4129  ωcom 4340  dom cdm 4372   ↾ cres 4374  ‘cfv 4929  recscrecs 5949  freccfrec 6007 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-id 4057  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-recs 5950  df-frec 6008 This theorem is referenced by:  frecsuclem3  6020
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