Theorem frrlem11 8246

Description: Lemma for well-founded recursion. For the next several theorems we will be aiming to prove that dom 𝐹 = 𝐴. To do this, we set up a function 𝐶 that supposedly contains an element of 𝐴 that is not in dom 𝐹 and we show that the element must be in dom 𝐹. Our choice of what to restrict 𝐹 to depends on if we assume partial orders or the axiom of infinity. To begin with, we establish the functionality of 𝐶. (Contributed by Scott Fenton, 7-Dec-2022.)

Hypotheses

Ref	Expression
frrlem11.1	⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
frrlem11.2	⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺)
frrlem11.3	⊢ ((𝜑 ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣))
frrlem11.4	⊢ 𝐶 = ((𝐹 ↾ 𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})

Assertion

Ref	Expression
frrlem11	⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 ∖ dom 𝐹)) → 𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧}))

Distinct variable groups:   𝐴,𝑓,𝑥,𝑦,𝑧   𝑓,𝐺,𝑥,𝑦,𝑧   𝑅,𝑓,𝑥,𝑦,𝑧   𝐵,𝑔,ℎ,𝑧   𝑥,𝐹,𝑢,𝑣,𝑧   𝜑,𝑓,𝑧   𝑓,𝐹   𝜑,𝑔,ℎ,𝑥,𝑢,𝑣

Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑣,𝑢,𝑔,ℎ)   𝐵(𝑥,𝑦,𝑣,𝑢,𝑓)   𝐶(𝑥,𝑦,𝑧,𝑣,𝑢,𝑓,𝑔,ℎ)   𝑅(𝑣,𝑢,𝑔,ℎ)   𝑆(𝑥,𝑦,𝑧,𝑣,𝑢,𝑓,𝑔,ℎ)   𝐹(𝑦,𝑔,ℎ)   𝐺(𝑣,𝑢,𝑔,ℎ)

Proof of Theorem frrlem11

Step	Hyp	Ref	Expression
1		frrlem11.1	. . . . . . 7 ⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
2		frrlem11.2	. . . . . . 7 ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺)
3		frrlem11.3	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣))
4	1, 2, 3	frrlem9 8244	. . . . . 6 ⊢ (𝜑 → Fun 𝐹)
5	4	funresd 6541	. . . . 5 ⊢ (𝜑 → Fun (𝐹 ↾ 𝑆))
6		dmres 5977	. . . . . 6 ⊢ dom (𝐹 ↾ 𝑆) = (𝑆 ∩ dom 𝐹)
7		df-fn 6501	. . . . . 6 ⊢ ((𝐹 ↾ 𝑆) Fn (𝑆 ∩ dom 𝐹) ↔ (Fun (𝐹 ↾ 𝑆) ∧ dom (𝐹 ↾ 𝑆) = (𝑆 ∩ dom 𝐹)))
8	6, 7	mpbiran2 711	. . . . 5 ⊢ ((𝐹 ↾ 𝑆) Fn (𝑆 ∩ dom 𝐹) ↔ Fun (𝐹 ↾ 𝑆))
9	5, 8	sylibr 234	. . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝑆) Fn (𝑆 ∩ dom 𝐹))
10		vex 3433	. . . . 5 ⊢ 𝑧 ∈ V
11		ovex 7400	. . . . 5 ⊢ (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∈ V
12	10, 11	fnsn 6556	. . . 4 ⊢ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} Fn {𝑧}
13	9, 12	jctir 520	. . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝑆) Fn (𝑆 ∩ dom 𝐹) ∧ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} Fn {𝑧}))
14		eldifn 4072	. . . . 5 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ dom 𝐹)
15		elinel2 4142	. . . . 5 ⊢ (𝑧 ∈ (𝑆 ∩ dom 𝐹) → 𝑧 ∈ dom 𝐹)
16	14, 15	nsyl 140	. . . 4 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ (𝑆 ∩ dom 𝐹))
17		disjsn 4655	. . . 4 ⊢ (((𝑆 ∩ dom 𝐹) ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ (𝑆 ∩ dom 𝐹))
18	16, 17	sylibr 234	. . 3 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ((𝑆 ∩ dom 𝐹) ∩ {𝑧}) = ∅)
19		fnun 6612	. . 3 ⊢ ((((𝐹 ↾ 𝑆) Fn (𝑆 ∩ dom 𝐹) ∧ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} Fn {𝑧}) ∧ ((𝑆 ∩ dom 𝐹) ∩ {𝑧}) = ∅) → ((𝐹 ↾ 𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧}))
20	13, 18, 19	syl2an 597	. 2 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 ∖ dom 𝐹)) → ((𝐹 ↾ 𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧}))
21		frrlem11.4	. . 3 ⊢ 𝐶 = ((𝐹 ↾ 𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
22	21	fneq1i 6595	. 2 ⊢ (𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ↔ ((𝐹 ↾ 𝑆) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧}))
23	20, 22	sylibr 234	1 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 ∖ dom 𝐹)) → 𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2714  ∀wral 3051   ∖ cdif 3886   ∪ cun 3887   ∩ cin 3888   ⊆ wss 3889  ∅c0 4273  {csn 4567  ⟨cop 4573   class class class wbr 5085  dom cdm 5631   ↾ cres 5633  Predcpred 6264  Fun wfun 6492   Fn wfn 6493  ‘cfv 6498  (class class class)co 7367  frecscfrecs 8230

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2539  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-iota 6454  df-fun 6500  df-fn 6501  df-fv 6506  df-ov 7370  df-frecs 8231

This theorem is referenced by:  frrlem12  8247  frrlem13  8248