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Theorem bnj900 32090
Description: Technical lemma for bnj69 32169. 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 32000 . . . . 5 (𝑓𝐵 → ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓))
3 simp1 1130 . . . . . 6 ((𝑓 Fn 𝑛𝜑𝜓) → 𝑓 Fn 𝑛)
43reximi 3248 . . . . 5 (∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) → ∃𝑛𝐷 𝑓 Fn 𝑛)
5 fndm 6452 . . . . . 6 (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
65reximi 3248 . . . . 5 (∃𝑛𝐷 𝑓 Fn 𝑛 → ∃𝑛𝐷 dom 𝑓 = 𝑛)
72, 4, 63syl 18 . . . 4 (𝑓𝐵 → ∃𝑛𝐷 dom 𝑓 = 𝑛)
87bnj1196 31955 . . 3 (𝑓𝐵 → ∃𝑛(𝑛𝐷 ∧ dom 𝑓 = 𝑛))
9 nfre1 3311 . . . . . . 7 𝑛𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)
109nfab 2989 . . . . . 6 𝑛{𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
111, 10nfcxfr 2980 . . . . 5 𝑛𝐵
1211nfcri 2976 . . . 4 𝑛 𝑓𝐵
131219.37 2227 . . 3 (∃𝑛(𝑓𝐵 → (𝑛𝐷 ∧ dom 𝑓 = 𝑛)) ↔ (𝑓𝐵 → ∃𝑛(𝑛𝐷 ∧ dom 𝑓 = 𝑛)))
148, 13mpbir 232 . 2 𝑛(𝑓𝐵 → (𝑛𝐷 ∧ dom 𝑓 = 𝑛))
15 nfv 1908 . . . 4 𝑛∅ ∈ dom 𝑓
1612, 15nfim 1890 . . 3 𝑛(𝑓𝐵 → ∅ ∈ dom 𝑓)
17 bnj900.3 . . . . . 6 𝐷 = (ω ∖ {∅})
1817bnj529 31901 . . . . 5 (𝑛𝐷 → ∅ ∈ 𝑛)
19 eleq2 2906 . . . . . 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 1081   = wceq 1530  wex 1773  wcel 2107  {cab 2804  wrex 3144  cdif 3937  c0 4295  {csn 4564  dom cdm 5554   Fn wfn 6347  ωcom 7568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-tr 5170  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-fn 6355  df-om 7569
This theorem is referenced by:  bnj906  32091
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