Theorem bnj1518 35741

Description: Technical lemma for bnj1500 35745. 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 1946	. . . 4 ⊢ Ⅎ𝑑𝜑
3		bnj1518.3	. . . . . 6 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
4		nfre1 3287	. . . . . . 7 ⊢ Ⅎ𝑑∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))
5	4	nfab 2928	. . . . . 6 ⊢ Ⅎ𝑑{𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
6	3, 5	nfcxfr 2920	. . . . 5 ⊢ Ⅎ𝑑𝐶
7	6	nfcri 2914	. . . 4 ⊢ Ⅎ𝑑 𝑓 ∈ 𝐶
8		nfv 1946	. . . 4 ⊢ Ⅎ𝑑 𝑥 ∈ dom 𝑓
9	2, 7, 8	nf3an 1933	. . 3 ⊢ Ⅎ𝑑(𝜑 ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓)
10	1, 9	nfxfr 1885	. 2 ⊢ Ⅎ𝑑𝜓
11	10	nf5ri 2231	1 ⊢ (𝜓 → ∀𝑑𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1567   = wceq 1569   ∈ wcel 2144  {cab 2738  ∀wral 3076  ∃wrex 3086   ⊆ wss 3895  ⟨cop 4834  ∪ cuni 4866  dom cdm 5650   ↾ cres 5652   Fn wfn 6534  ‘cfv 6539   predc-bnj14 35366   FrSe w-bnj15 35370

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-rex 3087

This theorem is used by:  bnj1501  35744