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Theorem frecsuclem1 5899
Description: Lemma for frecsuc 5903. (Contributed by Jim Kingdon, 13-Aug-2019.)
Hypothesis
Ref Expression
frecsuclem1.h 𝐺 = (g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})
Assertion
Ref Expression
frecsuclem1 ((𝐹 Fn V A 𝑉 B 𝜔) → (frec(𝐹, A)‘suc B) = (𝐺‘(recs(𝐺) ↾ suc B)))
Distinct variable groups:   g,𝑚,x,A   g,𝐹,x,𝑚   x,B   g,𝑉,𝑚   B,g,𝑚   g,𝐺,𝑚,x   x,𝑉

Proof of Theorem frecsuclem1
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 df-frec 5894 . . . . . 6 frec(𝐹, A) = (recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) ↾ 𝜔)
2 frecsuclem1.h . . . . . . . 8 𝐺 = (g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})
3 recseq 5839 . . . . . . . 8 (𝐺 = (g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))}) → recs(𝐺) = recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})))
42, 3ax-mp 7 . . . . . . 7 recs(𝐺) = recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))}))
54reseq1i 4531 . . . . . 6 (recs(𝐺) ↾ 𝜔) = (recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) ↾ 𝜔)
61, 5eqtr4i 2041 . . . . 5 frec(𝐹, A) = (recs(𝐺) ↾ 𝜔)
76fveq1i 5100 . . . 4 (frec(𝐹, A)‘suc B) = ((recs(𝐺) ↾ 𝜔)‘suc B)
8 peano2 4241 . . . . 5 (B 𝜔 → suc B 𝜔)
9 fvres 5119 . . . . 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 2062 . . 3 (B 𝜔 → (frec(𝐹, A)‘suc B) = (recs(𝐺)‘suc B))
12113ad2ant3 913 . 2 ((𝐹 Fn V A 𝑉 B 𝜔) → (frec(𝐹, A)‘suc B) = (recs(𝐺)‘suc B))
13 nnon 4255 . . . . 5 (suc B 𝜔 → suc B On)
148, 13syl 14 . . . 4 (B 𝜔 → suc B On)
15 eqid 2018 . . . . 5 recs(𝐺) = recs(𝐺)
162frectfr 5896 . . . . 5 ((𝐹 Fn V A 𝑉) → y(Fun 𝐺 (𝐺y) V))
1715, 16tfri2d 5868 . . . 4 (((𝐹 Fn V A 𝑉) suc B On) → (recs(𝐺)‘suc B) = (𝐺‘(recs(𝐺) ↾ suc B)))
1814, 17sylan2 270 . . 3 (((𝐹 Fn V A 𝑉) B 𝜔) → (recs(𝐺)‘suc B) = (𝐺‘(recs(𝐺) ↾ suc B)))
19183impa 1083 . 2 ((𝐹 Fn V A 𝑉 B 𝜔) → (recs(𝐺)‘suc B) = (𝐺‘(recs(𝐺) ↾ suc B)))
2012, 19eqtrd 2050 1 ((𝐹 Fn V A 𝑉 B 𝜔) → (frec(𝐹, A)‘suc B) = (𝐺‘(recs(𝐺) ↾ suc B)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wo 616   w3a 871   = wceq 1226   wcel 1370  {cab 2004  wrex 2281  Vcvv 2531  c0 3197  cmpt 3788  Oncon0 4045  suc csuc 4047  𝜔com 4236  dom cdm 4268  cres 4270   Fn wfn 4820  cfv 4825  recscrecs 5837  freccfrec 5893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-coll 3842  ax-sep 3845  ax-nul 3853  ax-pow 3897  ax-pr 3914  ax-un 4116  ax-setind 4200  ax-iinf 4234
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ne 2184  df-ral 2285  df-rex 2286  df-reu 2287  df-rab 2289  df-v 2533  df-sbc 2738  df-csb 2826  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-nul 3198  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-int 3586  df-iun 3629  df-br 3735  df-opab 3789  df-mpt 3790  df-tr 3825  df-id 4000  df-iord 4048  df-on 4050  df-suc 4053  df-iom 4237  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832  df-fv 4833  df-recs 5838  df-frec 5894
This theorem is referenced by:  frecsuclem3  5902
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