Theorem tfrlem10 8395

Description: Lemma for transfinite recursion. We define class 𝐶 by extending recs with one ordered pair. We will assume, falsely, that domain of recs is a member of, and thus not equal to, On. Using this assumption we will prove facts about 𝐶 that will lead to a contradiction in tfrlem14 8399, thus showing the domain of recs does in fact equal On. Here we show (under the false assumption) that 𝐶 is a function extending the domain of recs(𝐹) by one. (Contributed by NM, 14-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)

Hypotheses

Ref	Expression
tfrlem.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
tfrlem.3	⊢ 𝐶 = (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})

Assertion

Ref	Expression
tfrlem10	⊢ (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹))

Distinct variable groups:   𝑥,𝑓,𝑦,𝐶   𝑓,𝐹,𝑥,𝑦

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

Proof of Theorem tfrlem10

Step	Hyp	Ref	Expression
1		fvex 6898	. . . . . 6 ⊢ (𝐹‘recs(𝐹)) ∈ V
2		funsng 6591	. . . . . 6 ⊢ ((dom recs(𝐹) ∈ On ∧ (𝐹‘recs(𝐹)) ∈ V) → Fun {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
3	1, 2	mpan2 704	. . . . 5 ⊢ (dom recs(𝐹) ∈ On → Fun {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
4		tfrlem.1	. . . . . 6 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
5	4	tfrlem7 8391	. . . . 5 ⊢ Fun recs(𝐹)
6	3, 5	jctil 529	. . . 4 ⊢ (dom recs(𝐹) ∈ On → (Fun recs(𝐹) ∧ Fun {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}))
7	1	dmsnop 6217	. . . . . 6 ⊢ dom {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩} = {dom recs(𝐹)}
8	7	ineq2i 4163	. . . . 5 ⊢ (dom recs(𝐹) ∩ dom {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = (dom recs(𝐹) ∩ {dom recs(𝐹)})
9	4	tfrlem8 8392	. . . . . 6 ⊢ Ord dom recs(𝐹)
10		orddisj 6401	. . . . . 6 ⊢ (Ord dom recs(𝐹) → (dom recs(𝐹) ∩ {dom recs(𝐹)}) = ∅)
11	9, 10	ax-mp 5	. . . . 5 ⊢ (dom recs(𝐹) ∩ {dom recs(𝐹)}) = ∅
12	8, 11	eqtri 2784	. . . 4 ⊢ (dom recs(𝐹) ∩ dom {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = ∅
13		funun 6586	. . . 4 ⊢ (((Fun recs(𝐹) ∧ Fun {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∧ (dom recs(𝐹) ∩ dom {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = ∅) → Fun (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}))
14	6, 12, 13	sylancl 598	. . 3 ⊢ (dom recs(𝐹) ∈ On → Fun (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}))
15	7	uneq2i 4112	. . . 4 ⊢ (dom recs(𝐹) ∪ dom {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = (dom recs(𝐹) ∪ {dom recs(𝐹)})
16		dmun 5892	. . . 4 ⊢ dom (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = (dom recs(𝐹) ∪ dom {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
17		df-suc 6368	. . . 4 ⊢ suc dom recs(𝐹) = (dom recs(𝐹) ∪ {dom recs(𝐹)})
18	15, 16, 17	3eqtr4i 2794	. . 3 ⊢ dom (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = suc dom recs(𝐹)
19		df-fn 6541	. . 3 ⊢ ((recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) Fn suc dom recs(𝐹) ↔ (Fun (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) ∧ dom (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = suc dom recs(𝐹)))
20	14, 18, 19	sylanblrc 602	. 2 ⊢ (dom recs(𝐹) ∈ On → (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) Fn suc dom recs(𝐹))
21		tfrlem.3	. . 3 ⊢ 𝐶 = (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
22	21	fneq1i 6636	. 2 ⊢ (𝐶 Fn suc dom recs(𝐹) ↔ (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) Fn suc dom recs(𝐹))
23	20, 22	sylibr 237	1 ⊢ (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2739  ∀wral 3077  ∃wrex 3087  Vcvv 3451   ∪ cun 3897   ∩ cin 3898  ∅c0 4279  {csn 4584  ⟨cop 4590  dom cdm 5651   ↾ cres 5653  Ord word 6361  Oncon0 6362  suc csuc 6364  Fun wfun 6532   Fn wfn 6533  ‘cfv 6538  recscrecs 8378

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 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7751

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  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 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544  df-fv 6546  df-ov 7423  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379

This theorem is used by:  tfrlem11  8396  tfrlem12  8397  tfrlem13  8398