Theorem bnj1307 35565

Description: Technical lemma for bnj60 35604. 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
bnj1307.1	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1307.2	⊢ (𝑤 ∈ 𝐵 → ∀𝑥 𝑤 ∈ 𝐵)

Assertion

Ref	Expression
bnj1307	⊢ (𝑤 ∈ 𝐶 → ∀𝑥 𝑤 ∈ 𝐶)

Distinct variable groups:   𝑤,𝐵   𝑤,𝑑,𝑥   𝑥,𝑓

Allowed substitution hints:   𝐵(𝑥, 𝑓, 𝑑)   𝐶(𝑥, 𝑤, 𝑓, 𝑑)   𝐺(𝑥, 𝑤, 𝑓, 𝑑)   𝑌(𝑥, 𝑤, 𝑓, 𝑑)

Proof of Theorem bnj1307

Step	Hyp	Ref	Expression
1		bnj1307.1	. . 3 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
2		bnj1307.2	. . . . . 6 ⊢ (𝑤 ∈ 𝐵 → ∀𝑥 𝑤 ∈ 𝐵)
3	2	nfcii 2911	. . . . 5 ⊢ Ⅎ𝑥𝐵
4		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑥 𝑓 Fn 𝑑
5		nfra1 3286	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)
6	4, 5	nfan 1932	. . . . 5 ⊢ Ⅎ𝑥(𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))
7	3, 6	nfrexw 3310	. . . 4 ⊢ Ⅎ𝑥∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))
8	7	nfab 2928	. . 3 ⊢ Ⅎ𝑥{𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
9	1, 8	nfcxfr 2920	. 2 ⊢ Ⅎ𝑥𝐶
10	9	nfcrii 2917	1 ⊢ (𝑤 ∈ 𝐶 → ∀𝑥 𝑤 ∈ 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145  {cab 2738  ∀wral 3076  ∃wrex 3086   Fn wfn 6530  ‘cfv 6535

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-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087

This theorem is used by:  bnj1311  35566  bnj1373  35572  bnj1498  35603  bnj1525  35611  bnj1523  35613