Theorem tfr1 8388

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 2760	. . . 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 8374	. . 3 ⊢ Fun recs(𝐺)
3	1	tfrlem14 8382	. . 3 ⊢ dom recs(𝐺) = On
4		df-fn 6537	. . 3 ⊢ (recs(𝐺) Fn On ↔ (Fun recs(𝐺) ∧ dom recs(𝐺) = On))
5	2, 3, 4	mpbir2an 724	. 2 ⊢ recs(𝐺) Fn On
6		tfr.1	. . 3 ⊢ 𝐹 = recs(𝐺)
7	6	fneq1i 6631	. 2 ⊢ (𝐹 Fn On ↔ recs(𝐺) Fn On)
8	5, 7	mpbir 234	1 ⊢ 𝐹 Fn On

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  {cab 2738  ∀wral 3076  ∃wrex 3086  dom cdm 5655   ↾ cres 5657  Oncon0 6358  Fun wfun 6528   Fn wfn 6529  ‘cfv 6534  recscrecs 8361

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6300  df-ord 6361  df-on 6362  df-suc 6364  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7418  df-2nd 7989  df-frecs 8282  df-wrecs 8313  df-recs 8362

This theorem is used by:  tfr2  8389  tfr3  8390  recsfnon  8394  rdgfnon  8409  dfac8alem  10054  dfac12lem1  10168  dfac12lem2  10169  zorn2lem1  10520  zorn2lem2  10521  zorn2lem4  10523  zorn2lem5  10524  zorn2lem6  10525  zorn2lem7  10526  ttukeylem3  10535  ttukeylem5  10537  ttukeylem6  10538  madeval  28126  newval  28129  madef  28130  onvf1odlem3  35732  onvf1odlem4  35733  onvf1od  35734  dnnumch1  43899  dnnumch3lem  43901  dnnumch3  43902  aomclem6  43914