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Theorem tfr1 7357
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 2609 . . . 4 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
21tfrlem7 7343 . . 3 Fun recs(𝐺)
31tfrlem14 7351 . . 3 dom recs(𝐺) = On
4 df-fn 5793 . . 3 (recs(𝐺) Fn On ↔ (Fun recs(𝐺) ∧ dom recs(𝐺) = On))
52, 3, 4mpbir2an 956 . 2 recs(𝐺) Fn On
6 tfr.1 . . 3 𝐹 = recs(𝐺)
76fneq1i 5885 . 2 (𝐹 Fn On ↔ recs(𝐺) Fn On)
85, 7mpbir 219 1 𝐹 Fn On
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1474  {cab 2595  wral 2895  wrex 2896  dom cdm 5028  cres 5030  Oncon0 5626  Fun wfun 5784   Fn wfn 5785  cfv 5790  recscrecs 7331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-wrecs 7271  df-recs 7332
This theorem is referenced by:  tfr2  7358  tfr3  7359  recsfnon  7363  rdgfnon  7378  dfac8alem  8712  dfac12lem1  8825  dfac12lem2  8826  zorn2lem1  9178  zorn2lem2  9179  zorn2lem4  9181  zorn2lem5  9182  zorn2lem6  9183  zorn2lem7  9184  ttukeylem3  9193  ttukeylem5  9195  ttukeylem6  9196  dnnumch1  36428  dnnumch3lem  36430  dnnumch3  36431  aomclem6  36443
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