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Theorem frectfr 5896
Description: Lemma to connect transfinite recursion theorems with finite recursion. That is, given the conditions 𝐹 Fn V and A 𝑉 on frec(𝐹, A), we want to be able to apply tfri1d 5867 or tfri2d 5868, and this lemma lets us satisfy hypotheses of those theorems.

(Contributed by Jim Kingdon, 15-Aug-2019.)

Hypothesis
Ref Expression
frectfr.1 𝐺 = (g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})
Assertion
Ref Expression
frectfr ((𝐹 Fn V A 𝑉) → y(Fun 𝐺 (𝐺y) V))
Distinct variable groups:   g,𝑚,x,y,A   g,𝐹,𝑚,x,y   g,𝑉,𝑚,y
Allowed substitution hints:   𝐺(x,y,g,𝑚)   𝑉(x)

Proof of Theorem frectfr
StepHypRef Expression
1 vex 2534 . . . . . . . 8 g V
21a1i 9 . . . . . . 7 ((𝐹 Fn V A 𝑉) → g V)
3 ax-ia1 99 . . . . . . 7 ((𝐹 Fn V A 𝑉) → 𝐹 Fn V)
4 ax-ia2 100 . . . . . . 7 ((𝐹 Fn V A 𝑉) → A 𝑉)
52, 3, 4frecabex 5895 . . . . . 6 ((𝐹 Fn V A 𝑉) → {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))} V)
65ralrimivw 2367 . . . . 5 ((𝐹 Fn V A 𝑉) → g V {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))} V)
7 frectfr.1 . . . . . 6 𝐺 = (g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})
87fnmpt 4947 . . . . 5 (g V {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))} V → 𝐺 Fn V)
96, 8syl 14 . . . 4 ((𝐹 Fn V A 𝑉) → 𝐺 Fn V)
10 vex 2534 . . . 4 y V
11 funfvex 5113 . . . . 5 ((Fun 𝐺 y dom 𝐺) → (𝐺y) V)
1211funfni 4921 . . . 4 ((𝐺 Fn V y V) → (𝐺y) V)
139, 10, 12sylancl 394 . . 3 ((𝐹 Fn V A 𝑉) → (𝐺y) V)
147funmpt2 4861 . . 3 Fun 𝐺
1513, 14jctil 295 . 2 ((𝐹 Fn V A 𝑉) → (Fun 𝐺 (𝐺y) V))
1615alrimiv 1732 1 ((𝐹 Fn V A 𝑉) → y(Fun 𝐺 (𝐺y) V))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wo 616  wal 1224   = wceq 1226   wcel 1370  {cab 2004  wral 2280  wrex 2281  Vcvv 2531  c0 3197  cmpt 3788  suc csuc 4047  𝜔com 4236  dom cdm 4268  Fun wfun 4819   Fn wfn 4820  cfv 4825
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-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-pow 3897  ax-pr 3914  ax-un 4116  ax-iinf 4234
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  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-ral 2285  df-rex 2286  df-reu 2287  df-rab 2289  df-v 2533  df-sbc 2738  df-csb 2826  df-un 2895  df-in 2897  df-ss 2904  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-id 4000  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
This theorem is referenced by:  frecfnom  5897  frec0g  5898  frecsuclem1  5899  frecsuclemdm  5900  frecsuclem3  5902
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