Theorem wfrlem13OLD 8353

Description: Lemma for well-ordered recursion. From here through wfrlem16OLD 8356, we aim to prove that dom 𝐹 = 𝐴. We do this by supposing that there is an element 𝑧 of 𝐴 that is not in dom 𝐹. We then define 𝐶 by extending dom 𝐹 with the appropriate value at 𝑧. We then show that 𝑧 cannot be an 𝑅 minimal element of (𝐴 ∖ dom 𝐹), meaning that (𝐴 ∖ dom 𝐹) must be empty, so dom 𝐹 = 𝐴. Here, we show that 𝐶 is a function extending the domain of 𝐹 by one. Obsolete as of 18-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)

Hypotheses

Ref	Expression
wfrlem13OLD.1	⊢ 𝑅 We 𝐴
wfrlem13OLD.2	⊢ 𝑅 Se 𝐴
wfrlem13OLD.3	⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
wfrlem13OLD.4	⊢ 𝐶 = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})

Assertion

Ref	Expression
wfrlem13OLD	⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝐶 Fn (dom 𝐹 ∪ {𝑧}))

Distinct variable groups:   𝑧,𝐴   𝑧,𝐹   𝑧,𝑅

Allowed substitution hints:   𝐶(𝑧)   𝐺(𝑧)

Proof of Theorem wfrlem13OLD

Step	Hyp	Ref	Expression
1		wfrlem13OLD.1	. . . . 5 ⊢ 𝑅 We 𝐴
2		wfrlem13OLD.2	. . . . 5 ⊢ 𝑅 Se 𝐴
3		wfrlem13OLD.3	. . . . 5 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
4	1, 2, 3	wfrfunOLD 8351	. . . 4 ⊢ Fun 𝐹
5		vex 3466	. . . . 5 ⊢ 𝑧 ∈ V
6		fvex 6916	. . . . 5 ⊢ (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∈ V
7	5, 6	funsn 6614	. . . 4 ⊢ Fun {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}
8	4, 7	pm3.2i 469	. . 3 ⊢ (Fun 𝐹 ∧ Fun {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
9	6	dmsnop 6229	. . . . 5 ⊢ dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} = {𝑧}
10	9	ineq2i 4210	. . . 4 ⊢ (dom 𝐹 ∩ dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom 𝐹 ∩ {𝑧})
11		eldifn 4127	. . . . 5 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ dom 𝐹)
12		disjsn 4720	. . . . 5 ⊢ ((dom 𝐹 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ dom 𝐹)
13	11, 12	sylibr 233	. . . 4 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (dom 𝐹 ∩ {𝑧}) = ∅)
14	10, 13	eqtrid 2778	. . 3 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (dom 𝐹 ∩ dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = ∅)
15		funun 6607	. . 3 ⊢ (((Fun 𝐹 ∧ Fun {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) ∧ (dom 𝐹 ∩ dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = ∅) → Fun (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}))
16	8, 14, 15	sylancr 585	. 2 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → Fun (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}))
17		dmun 5919	. . 3 ⊢ dom (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom 𝐹 ∪ dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
18	9	uneq2i 4160	. . 3 ⊢ (dom 𝐹 ∪ dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom 𝐹 ∪ {𝑧})
19	17, 18	eqtri 2754	. 2 ⊢ dom (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom 𝐹 ∪ {𝑧})
20		wfrlem13OLD.4	. . . 4 ⊢ 𝐶 = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
21	20	fneq1i 6659	. . 3 ⊢ (𝐶 Fn (dom 𝐹 ∪ {𝑧}) ↔ (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) Fn (dom 𝐹 ∪ {𝑧}))
22		df-fn 6559	. . 3 ⊢ ((𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) Fn (dom 𝐹 ∪ {𝑧}) ↔ (Fun (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) ∧ dom (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom 𝐹 ∪ {𝑧})))
23	21, 22	bitri 274	. 2 ⊢ (𝐶 Fn (dom 𝐹 ∪ {𝑧}) ↔ (Fun (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) ∧ dom (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom 𝐹 ∪ {𝑧})))
24	16, 19, 23	sylanblrc 588	1 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝐶 Fn (dom 𝐹 ∪ {𝑧}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   = wceq 1534   ∈ wcel 2099   ∖ cdif 3944   ∪ cun 3945   ∩ cin 3946  ∅c0 4325  {csn 4633  ⟨cop 4639   Se wse 5637   We wwe 5638  dom cdm 5684   ↾ cres 5686  Predcpred 6313  Fun wfun 6550   Fn wfn 6551  ‘cfv 6556  wrecscwrecs 8328

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5306  ax-nul 5313  ax-pr 5435  ax-un 7748

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4916  df-iun 5005  df-br 5156  df-opab 5218  df-mpt 5239  df-id 5582  df-po 5596  df-so 5597  df-fr 5639  df-se 5640  df-we 5641  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6314  df-iota 6508  df-fun 6558  df-fn 6559  df-f 6560  df-fo 6562  df-fv 6564  df-ov 7429  df-2nd 8006  df-frecs 8298  df-wrecs 8329

This theorem is referenced by:  wfrlem14OLD  8354  wfrlem15OLD  8355