Theorem wfrlem13OLD 8344

Description: Obsolete version as of 18-Nov-2024. Lemma for well-ordered recursion. From here through wfrlem16OLD 8347, 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. (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 8342	. . . 4 ⊢ Fun 𝐹
5		vex 3468	. . . . 5 ⊢ 𝑧 ∈ V
6		fvex 6900	. . . . 5 ⊢ (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∈ V
7	5, 6	funsn 6600	. . . 4 ⊢ Fun {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}
8	4, 7	pm3.2i 470	. . 3 ⊢ (Fun 𝐹 ∧ Fun {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
9	6	dmsnop 6218	. . . . 5 ⊢ dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} = {𝑧}
10	9	ineq2i 4199	. . . 4 ⊢ (dom 𝐹 ∩ dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom 𝐹 ∩ {𝑧})
11		eldifn 4114	. . . . 5 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ dom 𝐹)
12		disjsn 4693	. . . . 5 ⊢ ((dom 𝐹 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ dom 𝐹)
13	11, 12	sylibr 234	. . . 4 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (dom 𝐹 ∩ {𝑧}) = ∅)
14	10, 13	eqtrid 2781	. . 3 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (dom 𝐹 ∩ dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = ∅)
15		funun 6593	. . 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 587	. 2 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → Fun (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}))
17		dmun 5903	. . 3 ⊢ dom (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom 𝐹 ∪ dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
18	9	uneq2i 4147	. . 3 ⊢ (dom 𝐹 ∪ dom {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = (dom 𝐹 ∪ {𝑧})
19	17, 18	eqtri 2757	. 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 6646	. . 3 ⊢ (𝐶 Fn (dom 𝐹 ∪ {𝑧}) ↔ (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) Fn (dom 𝐹 ∪ {𝑧}))
22		df-fn 6545	. . 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 275	. 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 590	1 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝐶 Fn (dom 𝐹 ∪ {𝑧}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107   ∖ cdif 3930   ∪ cun 3931   ∩ cin 3932  ∅c0 4315  {csn 4608  ⟨cop 4614   Se wse 5617   We wwe 5618  dom cdm 5667   ↾ cres 5669  Predcpred 6302  Fun wfun 6536   Fn wfn 6537  ‘cfv 6542  wrecscwrecs 8319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5278  ax-nul 5288  ax-pr 5414  ax-un 7738

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3421  df-v 3466  df-sbc 3773  df-csb 3882  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-iun 4975  df-br 5126  df-opab 5188  df-mpt 5208  df-id 5560  df-po 5574  df-so 5575  df-fr 5619  df-se 5620  df-we 5621  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6303  df-iota 6495  df-fun 6544  df-fn 6545  df-f 6546  df-fo 6548  df-fv 6550  df-ov 7417  df-2nd 7998  df-frecs 8289  df-wrecs 8320

This theorem is referenced by:  wfrlem14OLD  8345  wfrlem15OLD  8346