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Theorem frectfr 5985
 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 5949 or tfri2d 5950, 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 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})
Assertion
Ref Expression
frectfr ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → ∀𝑦(Fun 𝐺 ∧ (𝐺𝑦) ∈ V))
Distinct variable groups:   𝑔,𝑚,𝑥,𝑦,𝐴   𝑧,𝑔,𝐹,𝑚,𝑥,𝑦   𝑔,𝑉,𝑚,𝑦
Allowed substitution hints:   𝐴(𝑧)   𝐺(𝑥,𝑦,𝑧,𝑔,𝑚)   𝑉(𝑥,𝑧)

Proof of Theorem frectfr
StepHypRef Expression
1 vex 2560 . . . . . . . 8 𝑔 ∈ V
21a1i 9 . . . . . . 7 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → 𝑔 ∈ V)
3 simpl 102 . . . . . . 7 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → ∀𝑧(𝐹𝑧) ∈ V)
4 simpr 103 . . . . . . 7 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → 𝐴𝑉)
52, 3, 4frecabex 5984 . . . . . 6 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))} ∈ V)
65ralrimivw 2393 . . . . 5 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → ∀𝑔 ∈ V {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))} ∈ V)
7 frectfr.1 . . . . . 6 𝐺 = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})
87fnmpt 5025 . . . . 5 (∀𝑔 ∈ V {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))} ∈ V → 𝐺 Fn V)
96, 8syl 14 . . . 4 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → 𝐺 Fn V)
10 vex 2560 . . . 4 𝑦 ∈ V
11 funfvex 5192 . . . . 5 ((Fun 𝐺𝑦 ∈ dom 𝐺) → (𝐺𝑦) ∈ V)
1211funfni 4999 . . . 4 ((𝐺 Fn V ∧ 𝑦 ∈ V) → (𝐺𝑦) ∈ V)
139, 10, 12sylancl 392 . . 3 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → (𝐺𝑦) ∈ V)
147funmpt2 4939 . . 3 Fun 𝐺
1513, 14jctil 295 . 2 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → (Fun 𝐺 ∧ (𝐺𝑦) ∈ V))
1615alrimiv 1754 1 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉) → ∀𝑦(Fun 𝐺 ∧ (𝐺𝑦) ∈ V))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∨ wo 629  ∀wal 1241   = wceq 1243   ∈ wcel 1393  {cab 2026  ∀wral 2306  ∃wrex 2307  Vcvv 2557  ∅c0 3224   ↦ cmpt 3818  suc csuc 4102  ωcom 4313  dom cdm 4345  Fun wfun 4896   Fn wfn 4897  ‘cfv 4902 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-iinf 4311 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910 This theorem is referenced by:  frecfnom  5986  frecsuclem1  5987  frecsuclemdm  5988  frecsuclem3  5990
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