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Theorem tfri1 5838
 Description: Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of [TakeutiZaring] p. 47, with an additional condition. The condition is that 𝐺 is defined "everywhere" and here is stated as (𝐺‘x) ∈ V. Alternatively ∀x ∈ On∀f(f Fn x → f ∈ dom 𝐺) would suffice. Given a function 𝐺 satisfying that condition, we define a class A of all "acceptable" functions. The final function we're interested in is the union 𝐹 = recs(𝐺) of them. 𝐹 is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of 𝐹. In this first part we show that 𝐹 is a function whose domain is all ordinal numbers. (Contributed by Jim Kingdon, 4-May-2019.) (Revised by Mario Carneiro, 24-May-2019.)
Hypotheses
Ref Expression
tfri1.1 𝐹 = recs(𝐺)
tfri1.2 (Fun 𝐺 (𝐺x) V)
Assertion
Ref Expression
tfri1 𝐹 Fn On
Distinct variable group:   x,𝐺
Allowed substitution hint:   𝐹(x)

Proof of Theorem tfri1
Dummy variables f g u w y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2022 . . . 4 {wy On (w Fn y z y (wz) = (𝐺‘(wz)))} = {wy On (w Fn y z y (wz) = (𝐺‘(wz)))}
21tfrlem7 5820 . . 3 Fun recs(𝐺)
3 eqid 2022 . . . . 5 {gz On (g Fn z u z (gu) = (𝐺‘(gu)))} = {gz On (g Fn z u z (gu) = (𝐺‘(gu)))}
43tfrlem3 5813 . . . 4 {gz On (g Fn z u z (gu) = (𝐺‘(gu)))} = {fx On (f Fn x y x (fy) = (𝐺‘(fy)))}
5 tfri1.2 . . . 4 (Fun 𝐺 (𝐺x) V)
64, 5tfrlemi14 5834 . . 3 dom recs(𝐺) = On
7 df-fn 4799 . . 3 (recs(𝐺) Fn On ↔ (Fun recs(𝐺) dom recs(𝐺) = On))
82, 6, 7mpbir2an 837 . 2 recs(𝐺) Fn On
9 tfri1.1 . . 3 𝐹 = recs(𝐺)
109fneq1i 4886 . 2 (𝐹 Fn On ↔ recs(𝐺) Fn On)
118, 10mpbir 134 1 𝐹 Fn On
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1373   ∈ wcel 1375  {cab 2008  ∀wral 2282  ∃wrex 2283  Vcvv 2533  Oncon0 4024  dom cdm 4238   ↾ cres 4240  Fun wfun 4790   Fn wfn 4791  ‘cfv 4796  recscrecs 5806 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 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-13 1386  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004  ax-coll 3824  ax-sep 3827  ax-pow 3879  ax-pr 3896  ax-un 4093  ax-setind 4174 This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-fal 1232  df-nf 1329  df-sb 1628  df-eu 1884  df-mo 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2287  df-rex 2288  df-reu 2289  df-rab 2291  df-v 2535  df-sbc 2740  df-csb 2829  df-dif 2898  df-un 2900  df-in 2902  df-ss 2909  df-nul 3203  df-pw 3313  df-sn 3333  df-pr 3334  df-op 3336  df-uni 3533  df-iun 3611  df-br 3717  df-opab 3771  df-mpt 3772  df-tr 3807  df-id 3983  df-iord 4027  df-on 4028  df-suc 4031  df-xp 4244  df-rel 4245  df-cnv 4246  df-co 4247  df-dm 4248  df-rn 4249  df-res 4250  df-ima 4251  df-iota 4761  df-fun 4798  df-fn 4799  df-f 4800  df-f1 4801  df-fo 4802  df-f1o 4803  df-fv 4804  df-recs 5807 This theorem is referenced by:  tfri2  5839  tfri3  5840
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