Theorem bnj1518 35484

Description: Technical lemma for bnj1500 35488. 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
bnj1518.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1518.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1518.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1518.4	⊢ 𝐹 = ∪ 𝐶
bnj1518.5	⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴))
bnj1518.6	⊢ (𝜓 ↔ (𝜑 ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓))

Assertion

Ref	Expression
bnj1518	⊢ (𝜓 → ∀𝑑𝜓)

Distinct variable groups:   𝑓,𝑑   𝜑,𝑑   𝑥,𝑑

Allowed substitution hints:   𝜑(𝑥, 𝑓)   𝜓(𝑥, 𝑓, 𝑑)   𝐴(𝑥, 𝑓, 𝑑)   𝐵(𝑥, 𝑓, 𝑑)   𝐶(𝑥, 𝑓, 𝑑)   𝑅(𝑥, 𝑓, 𝑑)   𝐹(𝑥, 𝑓, 𝑑)   𝐺(𝑥, 𝑓, 𝑑)   𝑌(𝑥, 𝑓, 𝑑)

Proof of Theorem bnj1518

Step	Hyp	Ref	Expression
1		bnj1518.6	. . 3 ⊢ (𝜓 ↔ (𝜑 ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓))
2		nfv 1947	. . . 4 ⊢ Ⅎ𝑑𝜑
3		bnj1518.3	. . . . . 6 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
4		nfre1 3293	. . . . . . 7 ⊢ Ⅎ𝑑∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))
5	4	nfab 2934	. . . . . 6 ⊢ Ⅎ𝑑{𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
6	3, 5	nfcxfr 2926	. . . . 5 ⊢ Ⅎ𝑑𝐶
7	6	nfcri 2920	. . . 4 ⊢ Ⅎ𝑑 𝑓 ∈ 𝐶
8		nfv 1947	. . . 4 ⊢ Ⅎ𝑑 𝑥 ∈ dom 𝑓
9	2, 7, 8	nf3an 1934	. . 3 ⊢ Ⅎ𝑑(𝜑 ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓)
10	1, 9	nfxfr 1886	. 2 ⊢ Ⅎ𝑑𝜓
11	10	nf5ri 2234	1 ⊢ (𝜓 → ∀𝑑𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2146  {cab 2744  ∀wral 3082  ∃wrex 3092   ⊆ wss 3908  ⟨cop 4600  ∪ cuni 4877  dom cdm 5666   ↾ cres 5668   Fn wfn 6538  ‘cfv 6543   predc-bnj14 35109   FrSe w-bnj15 35113

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-rex 3093

This theorem is used by:  bnj1501  35487