Theorem tfrlem10 8443

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 8447, 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 6933	. . . . . 6 ⊢ (𝐹‘recs(𝐹)) ∈ V
2		funsng 6629	. . . . . 6 ⊢ ((dom recs(𝐹) ∈ On ∧ (𝐹‘recs(𝐹)) ∈ V) → Fun {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
3	1, 2	mpan2 690	. . . . 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 8439	. . . . 5 ⊢ Fun recs(𝐹)
6	3, 5	jctil 519	. . . 4 ⊢ (dom recs(𝐹) ∈ On → (Fun recs(𝐹) ∧ Fun {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}))
7	1	dmsnop 6247	. . . . . 6 ⊢ dom {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩} = {dom recs(𝐹)}
8	7	ineq2i 4238	. . . . 5 ⊢ (dom recs(𝐹) ∩ dom {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = (dom recs(𝐹) ∩ {dom recs(𝐹)})
9	4	tfrlem8 8440	. . . . . 6 ⊢ Ord dom recs(𝐹)
10		orddisj 6433	. . . . . 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 2768	. . . 4 ⊢ (dom recs(𝐹) ∩ dom {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = ∅
13		funun 6624	. . . 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 585	. . 3 ⊢ (dom recs(𝐹) ∈ On → Fun (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}))
15	7	uneq2i 4188	. . . 4 ⊢ (dom recs(𝐹) ∪ dom {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = (dom recs(𝐹) ∪ {dom recs(𝐹)})
16		dmun 5935	. . . 4 ⊢ dom (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = (dom recs(𝐹) ∪ dom {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
17		df-suc 6401	. . . 4 ⊢ suc dom recs(𝐹) = (dom recs(𝐹) ∪ {dom recs(𝐹)})
18	15, 16, 17	3eqtr4i 2778	. . 3 ⊢ dom (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = suc dom recs(𝐹)
19		df-fn 6576	. . 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 589	. 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 6676	. 2 ⊢ (𝐶 Fn suc dom recs(𝐹) ↔ (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) Fn suc dom recs(𝐹))
23	20, 22	sylibr 234	1 ⊢ (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cab 2717  ∀wral 3067  ∃wrex 3076  Vcvv 3488   ∪ cun 3974   ∩ cin 3975  ∅c0 4352  {csn 4648  ⟨cop 4654  dom cdm 5700   ↾ cres 5702  Ord word 6394  Oncon0 6395  suc csuc 6397  Fun wfun 6567   Fn wfn 6568  ‘cfv 6573  recscrecs 8426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fo 6579  df-fv 6581  df-ov 7451  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427

This theorem is referenced by:  tfrlem11  8444  tfrlem12  8445  tfrlem13  8446