Theorem bnj1245 35021

Description: Technical lemma for bnj60 35069. 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 34815	. . 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 35020	. . 3 ⊢ 𝐶 = 𝐾
8	2, 7	eleqtrdi 2841	. 2 ⊢ (𝜑 → ℎ ∈ 𝐾)
9	6	eqabri 2874	. . . . . 6 ⊢ (ℎ ∈ 𝐾 ↔ ∃𝑑 ∈ 𝐵 (ℎ Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘𝑍)))
10	9	bnj1238 34813	. . . . 5 ⊢ (ℎ ∈ 𝐾 → ∃𝑑 ∈ 𝐵 ℎ Fn 𝑑)
11	10	bnj1196 34801	. . . 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 2874	. . . . . 6 ⊢ (𝑑 ∈ 𝐵 ↔ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
14	13	simplbi 497	. . . . 5 ⊢ (𝑑 ∈ 𝐵 → 𝑑 ⊆ 𝐴)
15		fndm 6584	. . . . 5 ⊢ (ℎ Fn 𝑑 → dom ℎ = 𝑑)
16	14, 15	bnj1241 34814	. . . 4 ⊢ ((𝑑 ∈ 𝐵 ∧ ℎ Fn 𝑑) → dom ℎ ⊆ 𝐴)
17	11, 16	bnj593 34752	. . 3 ⊢ (ℎ ∈ 𝐾 → ∃𝑑dom ℎ ⊆ 𝐴)
18	17	bnj937 34778	. 2 ⊢ (ℎ ∈ 𝐾 → dom ℎ ⊆ 𝐴)
19	8, 18	syl 17	1 ⊢ (𝜑 → dom ℎ ⊆ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  {cab 2709   ≠ wne 2928  ∀wral 3047  ∃wrex 3056  {crab 3395   ∩ cin 3901   ⊆ wss 3902  ⟨cop 4582   class class class wbr 5091  dom cdm 5616   ↾ cres 5618   Fn wfn 6476  ‘cfv 6481   ∧ w-bnj17 34693   predc-bnj14 34695   FrSe w-bnj15 34699

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-res 5628  df-iota 6437  df-fun 6483  df-fn 6484  df-fv 6489  df-bnj17 34694

This theorem is referenced by: (None)