Theorem tfr1 7438

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 2621	. . . 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 7424	. . 3 ⊢ Fun recs(𝐺)
3	1	tfrlem14 7432	. . 3 ⊢ dom recs(𝐺) = On
4		df-fn 5850	. . 3 ⊢ (recs(𝐺) Fn On ↔ (Fun recs(𝐺) ∧ dom recs(𝐺) = On))
5	2, 3, 4	mpbir2an 954	. 2 ⊢ recs(𝐺) Fn On
6		tfr.1	. . 3 ⊢ 𝐹 = recs(𝐺)
7	6	fneq1i 5943	. 2 ⊢ (𝐹 Fn On ↔ recs(𝐺) Fn On)
8	5, 7	mpbir 221	1 ⊢ 𝐹 Fn On

Syntax hints:   ∧ wa 384   = wceq 1480  {cab 2607  ∀wral 2907  ∃wrex 2908  dom cdm 5074   ↾ cres 5076  Oncon0 5682  Fun wfun 5841   Fn wfn 5842  ‘cfv 5847  recscrecs 7412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-wrecs 7352  df-recs 7413

This theorem is referenced by:  tfr2  7439  tfr3  7440  recsfnon  7444  rdgfnon  7459  dfac8alem  8796  dfac12lem1  8909  dfac12lem2  8910  zorn2lem1  9262  zorn2lem2  9263  zorn2lem4  9265  zorn2lem5  9266  zorn2lem6  9267  zorn2lem7  9268  ttukeylem3  9277  ttukeylem5  9279  ttukeylem6  9280  dnnumch1  37091  dnnumch3lem  37093  dnnumch3  37094  aomclem6  37106