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Theorem tfr1 8022
Description: Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of [TakeutiZaring] p. 47. We start with an arbitrary class 𝐺, normally a function, and 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 NM, 17-Aug-1994.) (Revised by Mario Carneiro, 18-Jan-2015.)
Hypothesis
Ref Expression
tfr.1 𝐹 = recs(𝐺)
Assertion
Ref Expression
tfr1 𝐹 Fn On

Proof of Theorem tfr1
Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2818 . . . 4 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
21tfrlem7 8008 . . 3 Fun recs(𝐺)
31tfrlem14 8016 . . 3 dom recs(𝐺) = On
4 df-fn 6351 . . 3 (recs(𝐺) Fn On ↔ (Fun recs(𝐺) ∧ dom recs(𝐺) = On))
52, 3, 4mpbir2an 707 . 2 recs(𝐺) Fn On
6 tfr.1 . . 3 𝐹 = recs(𝐺)
76fneq1i 6443 . 2 (𝐹 Fn On ↔ recs(𝐺) Fn On)
85, 7mpbir 232 1 𝐹 Fn On
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1528  {cab 2796  wral 3135  wrex 3136  dom cdm 5548  cres 5550  Oncon0 6184  Fun wfun 6342   Fn wfn 6343  cfv 6348  recscrecs 7996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-wrecs 7936  df-recs 7997
This theorem is referenced by:  tfr2  8023  tfr3  8024  recsfnon  8028  rdgfnon  8043  dfac8alem  9443  dfac12lem1  9557  dfac12lem2  9558  zorn2lem1  9906  zorn2lem2  9907  zorn2lem4  9909  zorn2lem5  9910  zorn2lem6  9911  zorn2lem7  9912  ttukeylem3  9921  ttukeylem5  9923  ttukeylem6  9924  madeval  33186  dnnumch1  39522  dnnumch3lem  39524  dnnumch3  39525  aomclem6  39537
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