Theorem bnj1245 35627

Description: Technical lemma for bnj60 35675. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1245.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1245.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1245.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1245.4	⊢ 𝐷 = (dom 𝑔 ∩ dom ℎ)
bnj1245.5	⊢ 𝐸 = {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)}
bnj1245.6	⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)))
bnj1245.7	⊢ (𝜓 ↔ (𝜑 ∧ 𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥))
bnj1245.8	⊢ 𝑍 = ⟨𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1245.9	⊢ 𝐾 = {ℎ ∣ ∃𝑑 ∈ 𝐵 (ℎ Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘𝑍))}

Assertion

Ref	Expression
bnj1245	⊢ (𝜑 → dom ℎ ⊆ 𝐴)

Distinct variable groups:   𝐴,𝑑   𝐵,𝑓,ℎ   𝑓,𝐺,ℎ   ℎ,𝑌   𝑓,𝑍   𝑓,𝑑,ℎ   𝑥,𝑓,ℎ

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑓, 𝑔, ℎ, 𝑑)   𝜓(𝑥, 𝑦, 𝑓, 𝑔, ℎ, 𝑑)   𝐴(𝑥, 𝑦, 𝑓, 𝑔, ℎ)   𝐵(𝑥, 𝑦, 𝑔, 𝑑)   𝐶(𝑥, 𝑦, 𝑓, 𝑔, ℎ, 𝑑)   𝐷(𝑥, 𝑦, 𝑓, 𝑔, ℎ, 𝑑)   𝑅(𝑥, 𝑦, 𝑓, 𝑔, ℎ, 𝑑)   𝐸(𝑥, 𝑦, 𝑓, 𝑔, ℎ, 𝑑)   𝐺(𝑥, 𝑦, 𝑔, 𝑑)   𝐾(𝑥, 𝑦, 𝑓, 𝑔, ℎ, 𝑑)   𝑌(𝑥, 𝑦, 𝑓, 𝑔, 𝑑)   𝑍(𝑥, 𝑦, 𝑔, ℎ, 𝑑)

Proof of Theorem bnj1245

Step	Hyp	Ref	Expression
1		bnj1245.6	. . . 4 ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)))
2	1	bnj1247 35421	. . 3 ⊢ (𝜑 → ℎ ∈ 𝐶)
3		bnj1245.2	. . . 4 ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
4		bnj1245.3	. . . 4 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
5		bnj1245.8	. . . 4 ⊢ 𝑍 = ⟨𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))⟩
6		bnj1245.9	. . . 4 ⊢ 𝐾 = {ℎ ∣ ∃𝑑 ∈ 𝐵 (ℎ Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘𝑍))}
7	3, 4, 5, 6	bnj1234 35626	. . 3 ⊢ 𝐶 = 𝐾
8	2, 7	eleqtrdi 2871	. 2 ⊢ (𝜑 → ℎ ∈ 𝐾)
9	6	eqabri 2903	. . . . . 6 ⊢ (ℎ ∈ 𝐾 ↔ ∃𝑑 ∈ 𝐵 (ℎ Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘𝑍)))
10	9	bnj1238 35419	. . . . 5 ⊢ (ℎ ∈ 𝐾 → ∃𝑑 ∈ 𝐵 ℎ Fn 𝑑)
11	10	bnj1196 35407	. . . 4 ⊢ (ℎ ∈ 𝐾 → ∃𝑑(𝑑 ∈ 𝐵 ∧ ℎ Fn 𝑑))
12		bnj1245.1	. . . . . . 7 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
13	12	eqabri 2903	. . . . . 6 ⊢ (𝑑 ∈ 𝐵 ↔ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
14	13	simplbi 502	. . . . 5 ⊢ (𝑑 ∈ 𝐵 → 𝑑 ⊆ 𝐴)
15		fndm 6634	. . . . 5 ⊢ (ℎ Fn 𝑑 → dom ℎ = 𝑑)
16	14, 15	bnj1241 35420	. . . 4 ⊢ ((𝑑 ∈ 𝐵 ∧ ℎ Fn 𝑑) → dom ℎ ⊆ 𝐴)
17	11, 16	bnj593 35359	. . 3 ⊢ (ℎ ∈ 𝐾 → ∃𝑑dom ℎ ⊆ 𝐴)
18	17	bnj937 35385	. 2 ⊢ (ℎ ∈ 𝐾 → dom ℎ ⊆ 𝐴)
19	8, 18	syl 18	1 ⊢ (𝜑 → dom ℎ ⊆ 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {cab 2739   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  {crab 3413   ∩ cin 3898   ⊆ wss 3899  ⟨cop 4590   class class class wbr 5103  dom cdm 5651   ↾ cres 5653   Fn wfn 6526  ‘cfv 6531   ∧ w-bnj17 35300   predc-bnj14 35302   FrSe w-bnj15 35306

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-res 5663  df-iota 6487  df-fun 6533  df-fn 6534  df-fv 6539  df-bnj17 35301

This theorem is used by: (None)