Theorem bnj900 35126

Description: Technical lemma for bnj69 35207. 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 35036	. . . . 5 ⊢ (𝑓 ∈ 𝐵 → ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
3		simp1 1143	. . . . . 6 ⊢ ((𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → 𝑓 Fn 𝑛)
4	3	reximi 3079	. . . . 5 ⊢ (∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → ∃𝑛 ∈ 𝐷 𝑓 Fn 𝑛)
5		fndm 6592	. . . . . 6 ⊢ (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
6	5	reximi 3079	. . . . 5 ⊢ (∃𝑛 ∈ 𝐷 𝑓 Fn 𝑛 → ∃𝑛 ∈ 𝐷 dom 𝑓 = 𝑛)
7	2, 4, 6	3syl 18	. . . 4 ⊢ (𝑓 ∈ 𝐵 → ∃𝑛 ∈ 𝐷 dom 𝑓 = 𝑛)
8	7	bnj1196 34991	. . 3 ⊢ (𝑓 ∈ 𝐵 → ∃𝑛(𝑛 ∈ 𝐷 ∧ dom 𝑓 = 𝑛))
9		nfre1 3266	. . . . . . 7 ⊢ Ⅎ𝑛∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)
10	9	nfab 2909	. . . . . 6 ⊢ Ⅎ𝑛{𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
11	1, 10	nfcxfr 2901	. . . . 5 ⊢ Ⅎ𝑛𝐵
12	11	nfcri 2895	. . . 4 ⊢ Ⅎ𝑛 𝑓 ∈ 𝐵
13	12	19.37 2246	. . 3 ⊢ (∃𝑛(𝑓 ∈ 𝐵 → (𝑛 ∈ 𝐷 ∧ dom 𝑓 = 𝑛)) ↔ (𝑓 ∈ 𝐵 → ∃𝑛(𝑛 ∈ 𝐷 ∧ dom 𝑓 = 𝑛)))
14	8, 13	mpbir 233	. 2 ⊢ ∃𝑛(𝑓 ∈ 𝐵 → (𝑛 ∈ 𝐷 ∧ dom 𝑓 = 𝑛))
15		nfv 1922	. . . 4 ⊢ Ⅎ𝑛∅ ∈ dom 𝑓
16	12, 15	nfim 1904	. . 3 ⊢ Ⅎ𝑛(𝑓 ∈ 𝐵 → ∅ ∈ dom 𝑓)
17		bnj900.3	. . . . . 6 ⊢ 𝐷 = (ω ∖ {∅})
18	17	bnj529 34939	. . . . 5 ⊢ (𝑛 ∈ 𝐷 → ∅ ∈ 𝑛)
19		eleq2 2830	. . . . . 6 ⊢ (dom 𝑓 = 𝑛 → (∅ ∈ dom 𝑓 ↔ ∅ ∈ 𝑛))
20	19	biimparc 481	. . . . 5 ⊢ ((∅ ∈ 𝑛 ∧ dom 𝑓 = 𝑛) → ∅ ∈ dom 𝑓)
21	18, 20	sylan 587	. . . 4 ⊢ ((𝑛 ∈ 𝐷 ∧ dom 𝑓 = 𝑛) → ∅ ∈ dom 𝑓)
22	21	imim2i 16	. . 3 ⊢ ((𝑓 ∈ 𝐵 → (𝑛 ∈ 𝐷 ∧ dom 𝑓 = 𝑛)) → (𝑓 ∈ 𝐵 → ∅ ∈ dom 𝑓))
23	16, 22	exlimi 2231	. 2 ⊢ (∃𝑛(𝑓 ∈ 𝐵 → (𝑛 ∈ 𝐷 ∧ dom 𝑓 = 𝑛)) → (𝑓 ∈ 𝐵 → ∅ ∈ dom 𝑓))
24	14, 23	ax-mp 5	1 ⊢ (𝑓 ∈ 𝐵 → ∅ ∈ dom 𝑓)

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1093   = wceq 1548  ∃wex 1787   ∈ wcel 2121  {cab 2719  ∃wrex 3065   ∖ cdif 3882  ∅c0 4264  {csn 4558  dom cdm 5621   Fn wfn 6484  ωcom 7810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-tr 5183  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-ord 6317  df-on 6318  df-fn 6492  df-om 7811

This theorem is referenced by:  bnj906  35127