Theorem tfr1 8111

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.

Step	Hyp	Ref	Expression
1		eqid 2736	. . . 4 ⊢ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
2	1	tfrlem7 8097	. . 3 ⊢ Fun recs(𝐺)
3	1	tfrlem14 8105	. . 3 ⊢ dom recs(𝐺) = On
4		df-fn 6361	. . 3 ⊢ (recs(𝐺) Fn On ↔ (Fun recs(𝐺) ∧ dom recs(𝐺) = On))
5	2, 3, 4	mpbir2an 711	. 2 ⊢ recs(𝐺) Fn On
6		tfr.1	. . 3 ⊢ 𝐹 = recs(𝐺)
7	6	fneq1i 6454	. 2 ⊢ (𝐹 Fn On ↔ recs(𝐺) Fn On)
8	5, 7	mpbir 234	1 ⊢ 𝐹 Fn On

Syntax hints:   ∧ wa 399   = wceq 1543  {cab 2714  ∀wral 3051  ∃wrex 3052  dom cdm 5536   ↾ cres 5538  Oncon0 6191  Fun wfun 6352   Fn wfn 6353  ‘cfv 6358  recscrecs 8085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6140  df-ord 6194  df-on 6195  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-wrecs 8025  df-recs 8086

This theorem is referenced by:  tfr2  8112  tfr3  8113  recsfnon  8117  rdgfnon  8132  dfac8alem  9608  dfac12lem1  9722  dfac12lem2  9723  zorn2lem1  10075  zorn2lem2  10076  zorn2lem4  10078  zorn2lem5  10079  zorn2lem6  10080  zorn2lem7  10081  ttukeylem3  10090  ttukeylem5  10092  ttukeylem6  10093  madeval  33722  newval  33725  madef  33726  dnnumch1  40513  dnnumch3lem  40515  dnnumch3  40516  aomclem6  40528