Theorem tfrlem13 8385

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 8379	. . 3 ⊢ Ord dom recs(𝐹)
3		ordirr 6372	. . 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 2724	. . . . 5 ⊢ (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
6	1, 5	tfrlem12 8384	. . . 4 ⊢ (recs(𝐹) ∈ V → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ 𝐴)
7		elssuni 4931	. . . . 5 ⊢ ((recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ 𝐴 → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ⊆ ∪ 𝐴)
8	1	recsfval 8376	. . . . 5 ⊢ recs(𝐹) = ∪ 𝐴
9	7, 8	sseqtrrdi 4025	. . . 4 ⊢ ((recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ 𝐴 → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ⊆ recs(𝐹))
10		dmss 5892	. . . 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 7887	. . . . . 6 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ V)
14		elon2 6365	. . . . . 6 ⊢ (dom recs(𝐹) ∈ On ↔ (Ord dom recs(𝐹) ∧ dom recs(𝐹) ∈ V))
15	12, 13, 14	sylanbrc 582	. . . . 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 8382	. . . . 5 ⊢ (dom recs(𝐹) ∈ On → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) Fn suc dom recs(𝐹))
19		fndm 6642	. . . . 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 2828	. . 3 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ dom (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}))
22	11, 21	sseldd 3975	. 2 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ dom recs(𝐹))
23	4, 22	mto 196	1 ⊢ ¬ recs(𝐹) ∈ V

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {cab 2701  ∀wral 3053  ∃wrex 3062  Vcvv 3466   ∪ cun 3938   ⊆ wss 3940  {csn 4620  ⟨cop 4626  ∪ cuni 4899  dom cdm 5666   ↾ cres 5668  Ord word 6353  Oncon0 6354  suc csuc 6356   Fn wfn 6528  ‘cfv 6533  recscrecs 8365

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fo 6539  df-fv 6541  df-ov 7404  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366

This theorem is referenced by:  tfrlem14  8386  tfrlem15  8387  tfrlem16  8388  tfr2b  8391