Theorem bnj1307 35016

Description: Technical lemma for bnj60 35055. 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 2892	. . . . 5 ⊢ Ⅎ𝑥𝐵
4		nfv 1912	. . . . . 6 ⊢ Ⅎ𝑥 𝑓 Fn 𝑑
5		nfra1 3282	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)
6	4, 5	nfan 1897	. . . . 5 ⊢ Ⅎ𝑥(𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))
7	3, 6	nfrexw 3311	. . . 4 ⊢ Ⅎ𝑥∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))
8	7	nfab 2909	. . 3 ⊢ Ⅎ𝑥{𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
9	1, 8	nfcxfr 2901	. 2 ⊢ Ⅎ𝑥𝐶
10	9	nfcrii 2898	1 ⊢ (𝑤 ∈ 𝐶 → ∀𝑥 𝑤 ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1535   = wceq 1537   ∈ wcel 2106  {cab 2712  ∀wral 3059  ∃wrex 3068   Fn wfn 6558  ‘cfv 6563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069

This theorem is referenced by:  bnj1311  35017  bnj1373  35023  bnj1498  35054  bnj1525  35062  bnj1523  35064