Theorem tfr1 8368

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 2730	. . . 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 8354	. . 3 ⊢ Fun recs(𝐺)
3	1	tfrlem14 8362	. . 3 ⊢ dom recs(𝐺) = On
4		df-fn 6517	. . 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 6618	. 2 ⊢ (𝐹 Fn On ↔ recs(𝐺) Fn On)
8	5, 7	mpbir 231	1 ⊢ 𝐹 Fn On

Syntax hints:   ∧ wa 395   = wceq 1540  {cab 2708  ∀wral 3045  ∃wrex 3054  dom cdm 5641   ↾ cres 5643  Oncon0 6335  Fun wfun 6508   Fn wfn 6509  ‘cfv 6514  recscrecs 8342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343

This theorem is referenced by:  tfr2  8369  tfr3  8370  recsfnon  8374  rdgfnon  8389  dfac8alem  9989  dfac12lem1  10104  dfac12lem2  10105  zorn2lem1  10456  zorn2lem2  10457  zorn2lem4  10459  zorn2lem5  10460  zorn2lem6  10461  zorn2lem7  10462  ttukeylem3  10471  ttukeylem5  10473  ttukeylem6  10474  madeval  27767  newval  27770  madef  27771  onvf1odlem3  35099  onvf1odlem4  35100  onvf1od  35101  dnnumch1  43040  dnnumch3lem  43042  dnnumch3  43043  aomclem6  43055