Theorem bnj1259 35566

Description: Technical lemma for bnj60 35612. 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
bnj1259.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1259.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1259.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1259.4	⊢ 𝐷 = (dom 𝑔 ∩ dom ℎ)
bnj1259.5	⊢ 𝐸 = {𝑥 ∈ 𝐷 ∣ (𝑔‘𝑥) ≠ (ℎ‘𝑥)}
bnj1259.6	⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)))
bnj1259.7	⊢ (𝜓 ↔ (𝜑 ∧ 𝑥 ∈ 𝐸 ∧ ∀𝑦 ∈ 𝐸 ¬ 𝑦𝑅𝑥))

Assertion

Ref	Expression
bnj1259	⊢ (𝜑 → ∃𝑑 ∈ 𝐵 ℎ Fn 𝑑)

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓,ℎ   𝑓,𝐺,ℎ   𝑅,𝑓   ℎ,𝑌   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, 𝑔, ℎ, 𝑑)   𝐵(𝑥, 𝑦, 𝑔, 𝑑)   𝐶(𝑥, 𝑦, 𝑓, 𝑔, ℎ, 𝑑)   𝐷(𝑥, 𝑦, 𝑓, 𝑔, ℎ, 𝑑)   𝑅(𝑥, 𝑦, 𝑔, ℎ, 𝑑)   𝐸(𝑥, 𝑦, 𝑓, 𝑔, ℎ, 𝑑)   𝐺(𝑥, 𝑦, 𝑔, 𝑑)   𝑌(𝑥, 𝑦, 𝑓, 𝑔, 𝑑)

Proof of Theorem bnj1259

Step	Hyp	Ref	Expression
1		bnj1259.6	. 2 ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ∧ (𝑔 ↾ 𝐷) ≠ (ℎ ↾ 𝐷)))
2		abid 2742	. . . 4 ⊢ (ℎ ∈ {ℎ ∣ ∃𝑑 ∈ 𝐵 (ℎ Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘⟨𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))⟩))} ↔ ∃𝑑 ∈ 𝐵 (ℎ Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘⟨𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))⟩)))
3	2	bnj1238 35356	. . 3 ⊢ (ℎ ∈ {ℎ ∣ ∃𝑑 ∈ 𝐵 (ℎ Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘⟨𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))⟩))} → ∃𝑑 ∈ 𝐵 ℎ Fn 𝑑)
4		bnj1259.2	. . . 4 ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
5		bnj1259.3	. . . 4 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
6		eqid 2760	. . . 4 ⊢ ⟨𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))⟩
7		eqid 2760	. . . 4 ⊢ {ℎ ∣ ∃𝑑 ∈ 𝐵 (ℎ Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘⟨𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))⟩))} = {ℎ ∣ ∃𝑑 ∈ 𝐵 (ℎ Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘⟨𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))⟩))}
8	4, 5, 6, 7	bnj1234 35563	. . 3 ⊢ 𝐶 = {ℎ ∣ ∃𝑑 ∈ 𝐵 (ℎ Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (ℎ‘𝑥) = (𝐺‘⟨𝑥, (ℎ ↾ pred(𝑥, 𝐴, 𝑅))⟩))}
9	3, 8	eleq2s 2878	. 2 ⊢ (ℎ ∈ 𝐶 → ∃𝑑 ∈ 𝐵 ℎ Fn 𝑑)
10	1, 9	bnj771 35315	1 ⊢ (𝜑 → ∃𝑑 ∈ 𝐵 ℎ Fn 𝑑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {cab 2738   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412   ∩ cin 3898   ⊆ wss 3899  ⟨cop 4590   class class class wbr 5103  dom cdm 5648   ↾ cres 5650   Fn wfn 6523  ‘cfv 6528   ∧ w-bnj17 35237   predc-bnj14 35239   FrSe w-bnj15 35243

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 2732

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 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  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 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-res 5660  df-iota 6484  df-fun 6530  df-fn 6531  df-fv 6536  df-bnj17 35238

This theorem is used by:  bnj1253  35567  bnj1286  35569  bnj1280  35570