Theorem tfrlem10 8390

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 8394, 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 6905	. . . . . 6 ⊢ (𝐹‘recs(𝐹)) ∈ V
2		funsng 6600	. . . . . 6 ⊢ ((dom recs(𝐹) ∈ On ∧ (𝐹‘recs(𝐹)) ∈ V) → Fun {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
3	1, 2	mpan2 688	. . . . 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 8386	. . . . 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 6216	. . . . . 6 ⊢ dom {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩} = {dom recs(𝐹)}
8	7	ineq2i 4210	. . . . 5 ⊢ (dom recs(𝐹) ∩ dom {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = (dom recs(𝐹) ∩ {dom recs(𝐹)})
9	4	tfrlem8 8387	. . . . . 6 ⊢ Ord dom recs(𝐹)
10		orddisj 6403	. . . . . 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 2759	. . . 4 ⊢ (dom recs(𝐹) ∩ dom {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = ∅
13		funun 6595	. . . 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 4161	. . . 4 ⊢ (dom recs(𝐹) ∪ dom {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = (dom recs(𝐹) ∪ {dom recs(𝐹)})
16		dmun 5911	. . . 4 ⊢ dom (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = (dom recs(𝐹) ∪ dom {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})
17		df-suc 6371	. . . 4 ⊢ suc dom recs(𝐹) = (dom recs(𝐹) ∪ {dom recs(𝐹)})
18	15, 16, 17	3eqtr4i 2769	. . 3 ⊢ dom (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = suc dom recs(𝐹)
19		df-fn 6547	. . 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 6647	. 2 ⊢ (𝐶 Fn suc dom recs(𝐹) ↔ (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) Fn suc dom recs(𝐹))
23	20, 22	sylibr 233	1 ⊢ (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  {cab 2708  ∀wral 3060  ∃wrex 3069  Vcvv 3473   ∪ cun 3947   ∩ cin 3948  ∅c0 4323  {csn 4629  ⟨cop 4635  dom cdm 5677   ↾ cres 5679  Ord word 6364  Oncon0 6365  suc csuc 6367  Fun wfun 6538   Fn wfn 6539  ‘cfv 6544  recscrecs 8373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7728

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-ov 7415  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374

This theorem is referenced by:  tfrlem11  8391  tfrlem12  8392  tfrlem13  8393