Theorem tfrlem13 8361

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 8355	. . 3 ⊢ Ord dom recs(𝐹)
3		ordirr 6353	. . 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 2730	. . . . 5 ⊢ (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
6	1, 5	tfrlem12 8360	. . . 4 ⊢ (recs(𝐹) ∈ V → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ 𝐴)
7		elssuni 4904	. . . . 5 ⊢ ((recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ 𝐴 → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ⊆ ∪ 𝐴)
8	1	recsfval 8352	. . . . 5 ⊢ recs(𝐹) = ∪ 𝐴
9	7, 8	sseqtrrdi 3991	. . . 4 ⊢ ((recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∈ 𝐴 → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ⊆ recs(𝐹))
10		dmss 5869	. . . 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 7880	. . . . . 6 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ V)
14		elon2 6346	. . . . . 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 6418	. . . . 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 8358	. . . . 5 ⊢ (dom recs(𝐹) ∈ On → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) Fn suc dom recs(𝐹))
19		fndm 6624	. . . . 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 2832	. . 3 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ dom (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}))
22	11, 21	sseldd 3950	. 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 2109  {cab 2708  ∀wral 3045  ∃wrex 3054  Vcvv 3450   ∪ cun 3915   ⊆ wss 3917  {csn 4592  ⟨cop 4598  ∪ cuni 4874  dom cdm 5641   ↾ cres 5643  Ord word 6334  Oncon0 6335  suc csuc 6337   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-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-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-fo 6520  df-fv 6522  df-ov 7393  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343

This theorem is referenced by:  tfrlem14  8362  tfrlem15  8363  tfrlem16  8364  tfr2b  8367