Theorem tfrlem13 8404

Description: Lemma for transfinite recursion. If recs is a set function, then 𝐶 is acceptable, and thus a subset of recs, but dom 𝐶 is bigger than dom recs. This is a contradiction, so recs must be a proper class function. (Contributed by NM, 14-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2014.)

Hypothesis

Ref	Expression
tfrlem.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}

Assertion

Ref	Expression
tfrlem13	⊢ ¬ recs(𝐹) ∈ V

Distinct variable group:   𝑥,𝑓,𝑦,𝐹

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑓)

Proof of Theorem tfrlem13

Step	Hyp	Ref	Expression
1		tfrlem.1	. . . 4 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
2	1	tfrlem8 8398	. . 3 ⊢ Ord dom recs(𝐹)
3		ordirr 6370	. . 3 ⊢ (Ord dom recs(𝐹) → ¬ dom recs(𝐹) ∈ dom recs(𝐹))
4	2, 3	ax-mp 5	. 2 ⊢ ¬ dom recs(𝐹) ∈ dom recs(𝐹)
5		eqid 2735	. . . . 5 ⊢ (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
6	1, 5	tfrlem12 8403	. . . 4 ⊢ (recs(𝐹) ∈ V → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ 𝐴)
7		elssuni 4913	. . . . 5 ⊢ ((recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ 𝐴 → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ⊆ ∪ 𝐴)
8	1	recsfval 8395	. . . . 5 ⊢ recs(𝐹) = ∪ 𝐴
9	7, 8	sseqtrrdi 4000	. . . 4 ⊢ ((recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ 𝐴 → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ⊆ recs(𝐹))
10		dmss 5882	. . . 4 ⊢ ((recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ⊆ recs(𝐹) → dom (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ⊆ dom recs(𝐹))
11	6, 9, 10	3syl 18	. . 3 ⊢ (recs(𝐹) ∈ V → dom (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ⊆ dom recs(𝐹))
12	2	a1i 11	. . . . . 6 ⊢ (recs(𝐹) ∈ V → Ord dom recs(𝐹))
13		dmexg 7897	. . . . . 6 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ V)
14		elon2 6363	. . . . . 6 ⊢ (dom recs(𝐹) ∈ On ↔ (Ord dom recs(𝐹) ∧ dom recs(𝐹) ∈ V))
15	12, 13, 14	sylanbrc 583	. . . . 5 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ On)
16		sucidg 6435	. . . . 5 ⊢ (dom recs(𝐹) ∈ On → dom recs(𝐹) ∈ suc dom recs(𝐹))
17	15, 16	syl 17	. . . 4 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ suc dom recs(𝐹))
18	1, 5	tfrlem10 8401	. . . . 5 ⊢ (dom recs(𝐹) ∈ On → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) Fn suc dom recs(𝐹))
19		fndm 6641	. . . . 5 ⊢ ((recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) Fn suc dom recs(𝐹) → dom (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = suc dom recs(𝐹))
20	15, 18, 19	3syl 18	. . . 4 ⊢ (recs(𝐹) ∈ V → dom (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = suc dom recs(𝐹))
21	17, 20	eleqtrrd 2837	. . 3 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ dom (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}))
22	11, 21	sseldd 3959	. 2 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ dom recs(𝐹))
23	4, 22	mto 197	1 ⊢ ¬ recs(𝐹) ∈ V

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2713  ∀wral 3051  ∃wrex 3060  Vcvv 3459   ∪ cun 3924   ⊆ wss 3926  {csn 4601  ⟨cop 4607  ∪ cuni 4883  dom cdm 5654   ↾ cres 5656  Ord word 6351  Oncon0 6352  suc csuc 6354   Fn wfn 6526  ‘cfv 6531  recscrecs 8384

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fo 6537  df-fv 6539  df-ov 7408  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385

This theorem is referenced by:  tfrlem14  8405  tfrlem15  8406  tfrlem16  8407  tfr2b  8410