Theorem bnj1307 32903

Description: Technical lemma for bnj60 32942. 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 2890	. . . . 5 ⊢ Ⅎ𝑥𝐵
4		nfv 1918	. . . . . 6 ⊢ Ⅎ𝑥 𝑓 Fn 𝑑
5		nfra1 3142	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)
6	4, 5	nfan 1903	. . . . 5 ⊢ Ⅎ𝑥(𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))
7	3, 6	nfrex 3237	. . . 4 ⊢ Ⅎ𝑥∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))
8	7	nfab 2912	. . 3 ⊢ Ⅎ𝑥{𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
9	1, 8	nfcxfr 2904	. 2 ⊢ Ⅎ𝑥𝐶
10	9	nfcrii 2898	1 ⊢ (𝑤 ∈ 𝐶 → ∀𝑥 𝑤 ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1537   = wceq 1539   ∈ wcel 2108  {cab 2715  ∀wral 3063  ∃wrex 3064   Fn wfn 6413  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069

This theorem is referenced by:  bnj1311  32904  bnj1373  32910  bnj1498  32941  bnj1525  32949  bnj1523  32951