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Theorem bnj900 32909
Description: Technical lemma for bnj69 32990. 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 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
Assertion
Ref Expression
bnj900 (𝑓𝐵 → ∅ ∈ 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
StepHypRef Expression
1 bnj900.4 . . . . . 6 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
21bnj1436 32819 . . . . 5 (𝑓𝐵 → ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓))
3 simp1 1135 . . . . . 6 ((𝑓 Fn 𝑛𝜑𝜓) → 𝑓 Fn 𝑛)
43reximi 3178 . . . . 5 (∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) → ∃𝑛𝐷 𝑓 Fn 𝑛)
5 fndm 6536 . . . . . 6 (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
65reximi 3178 . . . . 5 (∃𝑛𝐷 𝑓 Fn 𝑛 → ∃𝑛𝐷 dom 𝑓 = 𝑛)
72, 4, 63syl 18 . . . 4 (𝑓𝐵 → ∃𝑛𝐷 dom 𝑓 = 𝑛)
87bnj1196 32774 . . 3 (𝑓𝐵 → ∃𝑛(𝑛𝐷 ∧ dom 𝑓 = 𝑛))
9 nfre1 3239 . . . . . . 7 𝑛𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)
109nfab 2913 . . . . . 6 𝑛{𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
111, 10nfcxfr 2905 . . . . 5 𝑛𝐵
1211nfcri 2894 . . . 4 𝑛 𝑓𝐵
131219.37 2225 . . 3 (∃𝑛(𝑓𝐵 → (𝑛𝐷 ∧ dom 𝑓 = 𝑛)) ↔ (𝑓𝐵 → ∃𝑛(𝑛𝐷 ∧ dom 𝑓 = 𝑛)))
148, 13mpbir 230 . 2 𝑛(𝑓𝐵 → (𝑛𝐷 ∧ dom 𝑓 = 𝑛))
15 nfv 1917 . . . 4 𝑛∅ ∈ dom 𝑓
1612, 15nfim 1899 . . 3 𝑛(𝑓𝐵 → ∅ ∈ dom 𝑓)
17 bnj900.3 . . . . . 6 𝐷 = (ω ∖ {∅})
1817bnj529 32721 . . . . 5 (𝑛𝐷 → ∅ ∈ 𝑛)
19 eleq2 2827 . . . . . 6 (dom 𝑓 = 𝑛 → (∅ ∈ dom 𝑓 ↔ ∅ ∈ 𝑛))
2019biimparc 480 . . . . 5 ((∅ ∈ 𝑛 ∧ dom 𝑓 = 𝑛) → ∅ ∈ dom 𝑓)
2118, 20sylan 580 . . . 4 ((𝑛𝐷 ∧ dom 𝑓 = 𝑛) → ∅ ∈ dom 𝑓)
2221imim2i 16 . . 3 ((𝑓𝐵 → (𝑛𝐷 ∧ dom 𝑓 = 𝑛)) → (𝑓𝐵 → ∅ ∈ dom 𝑓))
2316, 22exlimi 2210 . 2 (∃𝑛(𝑓𝐵 → (𝑛𝐷 ∧ dom 𝑓 = 𝑛)) → (𝑓𝐵 → ∅ ∈ dom 𝑓))
2414, 23ax-mp 5 1 (𝑓𝐵 → ∅ ∈ dom 𝑓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wex 1782  wcel 2106  {cab 2715  wrex 3065  cdif 3884  c0 4256  {csn 4561  dom cdm 5589   Fn wfn 6428  ωcom 7712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-ord 6269  df-on 6270  df-fn 6436  df-om 7713
This theorem is referenced by:  bnj906  32910
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