Theorem bnj900 35626

Description: Technical lemma for bnj69 35707. 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
bnj900.3	⊢ 𝐷 = (ω ∖ {∅})
bnj900.4	⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}

Assertion

Ref	Expression
bnj900	⊢ (𝑓 ∈ 𝐵 → ∅ ∈ dom 𝑓)

Distinct variable group:   𝑓,𝑛

Allowed substitution hints:   𝜑(𝑓, 𝑛)   𝜓(𝑓, 𝑛)   𝐵(𝑓, 𝑛)   𝐷(𝑓, 𝑛)

Proof of Theorem bnj900

Step	Hyp	Ref	Expression
1		bnj900.4	. . . . . 6 ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
2	1	bnj1436 35536	. . . . 5 ⊢ (𝑓 ∈ 𝐵 → ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
3		simp1 1154	. . . . . 6 ⊢ ((𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → 𝑓 Fn 𝑛)
4	3	reximi 3104	. . . . 5 ⊢ (∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → ∃𝑛 ∈ 𝐷 𝑓 Fn 𝑛)
5		fndm 6661	. . . . . 6 ⊢ (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
6	5	reximi 3104	. . . . 5 ⊢ (∃𝑛 ∈ 𝐷 𝑓 Fn 𝑛 → ∃𝑛 ∈ 𝐷 dom 𝑓 = 𝑛)
7	2, 4, 6	3syl 19	. . . 4 ⊢ (𝑓 ∈ 𝐵 → ∃𝑛 ∈ 𝐷 dom 𝑓 = 𝑛)
8	7	bnj1196 35491	. . 3 ⊢ (𝑓 ∈ 𝐵 → ∃𝑛(𝑛 ∈ 𝐷 ∧ dom 𝑓 = 𝑛))
9		nfre1 3291	. . . . . . 7 ⊢ Ⅎ𝑛∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)
10	9	nfab 2932	. . . . . 6 ⊢ Ⅎ𝑛{𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
11	1, 10	nfcxfr 2924	. . . . 5 ⊢ Ⅎ𝑛𝐵
12	11	nfcri 2918	. . . 4 ⊢ Ⅎ𝑛 𝑓 ∈ 𝐵
13	12	19.37 2272	. . 3 ⊢ (∃𝑛(𝑓 ∈ 𝐵 → (𝑛 ∈ 𝐷 ∧ dom 𝑓 = 𝑛)) ↔ (𝑓 ∈ 𝐵 → ∃𝑛(𝑛 ∈ 𝐷 ∧ dom 𝑓 = 𝑛)))
14	8, 13	mpbir 234	. 2 ⊢ ∃𝑛(𝑓 ∈ 𝐵 → (𝑛 ∈ 𝐷 ∧ dom 𝑓 = 𝑛))
15		nfv 1949	. . . 4 ⊢ Ⅎ𝑛∅ ∈ dom 𝑓
16	12, 15	nfim 1931	. . 3 ⊢ Ⅎ𝑛(𝑓 ∈ 𝐵 → ∅ ∈ dom 𝑓)
17		bnj900.3	. . . . . 6 ⊢ 𝐷 = (ω ∖ {∅})
18	17	bnj529 35439	. . . . 5 ⊢ (𝑛 ∈ 𝐷 → ∅ ∈ 𝑛)
19		eleq2 2853	. . . . . 6 ⊢ (dom 𝑓 = 𝑛 → (∅ ∈ dom 𝑓 ↔ ∅ ∈ 𝑛))
20	19	biimparc 485	. . . . 5 ⊢ ((∅ ∈ 𝑛 ∧ dom 𝑓 = 𝑛) → ∅ ∈ dom 𝑓)
21	18, 20	sylan 592	. . . 4 ⊢ ((𝑛 ∈ 𝐷 ∧ dom 𝑓 = 𝑛) → ∅ ∈ dom 𝑓)
22	21	imim2i 17	. . 3 ⊢ ((𝑓 ∈ 𝐵 → (𝑛 ∈ 𝐷 ∧ dom 𝑓 = 𝑛)) → (𝑓 ∈ 𝐵 → ∅ ∈ dom 𝑓))
23	16, 22	exlimi 2257	. 2 ⊢ (∃𝑛(𝑓 ∈ 𝐵 → (𝑛 ∈ 𝐷 ∧ dom 𝑓 = 𝑛)) → (𝑓 ∈ 𝐵 → ∅ ∈ dom 𝑓))
24	14, 23	ax-mp 5	1 ⊢ (𝑓 ∈ 𝐵 → ∅ ∈ dom 𝑓)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  ∃wrex 3090   ∖ cdif 4048  ∅c0 4281  {csn 4588  dom cdm 5655   Fn wfn 6550  ωcom 7898

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-tr 5216  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-ord 6380  df-on 6381  df-fn 6558  df-om 7899

This theorem is used by:  bnj906  35627