Theorem tfr1 8340

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 2737	. . . 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 8326	. . 3 ⊢ Fun recs(𝐺)
3	1	tfrlem14 8334	. . 3 ⊢ dom recs(𝐺) = On
4		df-fn 6505	. . 3 ⊢ (recs(𝐺) Fn On ↔ (Fun recs(𝐺) ∧ dom recs(𝐺) = On))
5	2, 3, 4	mpbir2an 712	. 2 ⊢ recs(𝐺) Fn On
6		tfr.1	. . 3 ⊢ 𝐹 = recs(𝐺)
7	6	fneq1i 6599	. 2 ⊢ (𝐹 Fn On ↔ recs(𝐺) Fn On)
8	5, 7	mpbir 231	1 ⊢ 𝐹 Fn On

Syntax hints:   ∧ wa 395   = wceq 1542  {cab 2715  ∀wral 3052  ∃wrex 3062  dom cdm 5634   ↾ cres 5636  Oncon0 6327  Fun wfun 6496   Fn wfn 6497  ‘cfv 6502  recscrecs 8314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-ov 7373  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315

This theorem is referenced by:  tfr2  8341  tfr3  8342  recsfnon  8346  rdgfnon  8361  dfac8alem  9953  dfac12lem1  10068  dfac12lem2  10069  zorn2lem1  10420  zorn2lem2  10421  zorn2lem4  10423  zorn2lem5  10424  zorn2lem6  10425  zorn2lem7  10426  ttukeylem3  10435  ttukeylem5  10437  ttukeylem6  10438  madeval  27845  newval  27848  madef  27849  onvf1odlem3  35327  onvf1odlem4  35328  onvf1od  35329  dnnumch1  43430  dnnumch3lem  43432  dnnumch3  43433  aomclem6  43445