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Theorem bnj900 31125
Description: Technical lemma for bnj69 31204. 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 31036 . . . . 5 (𝑓𝐵 → ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓))
3 simp1 1081 . . . . . 6 ((𝑓 Fn 𝑛𝜑𝜓) → 𝑓 Fn 𝑛)
43reximi 3040 . . . . 5 (∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) → ∃𝑛𝐷 𝑓 Fn 𝑛)
5 fndm 6028 . . . . . 6 (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
65reximi 3040 . . . . 5 (∃𝑛𝐷 𝑓 Fn 𝑛 → ∃𝑛𝐷 dom 𝑓 = 𝑛)
72, 4, 63syl 18 . . . 4 (𝑓𝐵 → ∃𝑛𝐷 dom 𝑓 = 𝑛)
87bnj1196 30991 . . 3 (𝑓𝐵 → ∃𝑛(𝑛𝐷 ∧ dom 𝑓 = 𝑛))
9 nfre1 3034 . . . . . . 7 𝑛𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)
109nfab 2798 . . . . . 6 𝑛{𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
111, 10nfcxfr 2791 . . . . 5 𝑛𝐵
1211nfcri 2787 . . . 4 𝑛 𝑓𝐵
131219.37 2138 . . 3 (∃𝑛(𝑓𝐵 → (𝑛𝐷 ∧ dom 𝑓 = 𝑛)) ↔ (𝑓𝐵 → ∃𝑛(𝑛𝐷 ∧ dom 𝑓 = 𝑛)))
148, 13mpbir 221 . 2 𝑛(𝑓𝐵 → (𝑛𝐷 ∧ dom 𝑓 = 𝑛))
15 nfv 1883 . . . 4 𝑛∅ ∈ dom 𝑓
1612, 15nfim 1865 . . 3 𝑛(𝑓𝐵 → ∅ ∈ dom 𝑓)
17 bnj900.3 . . . . . 6 𝐷 = (ω ∖ {∅})
1817bnj529 30937 . . . . 5 (𝑛𝐷 → ∅ ∈ 𝑛)
19 eleq2 2719 . . . . . 6 (dom 𝑓 = 𝑛 → (∅ ∈ dom 𝑓 ↔ ∅ ∈ 𝑛))
2019biimparc 503 . . . . 5 ((∅ ∈ 𝑛 ∧ dom 𝑓 = 𝑛) → ∅ ∈ dom 𝑓)
2118, 20sylan 487 . . . 4 ((𝑛𝐷 ∧ dom 𝑓 = 𝑛) → ∅ ∈ dom 𝑓)
2221imim2i 16 . . 3 ((𝑓𝐵 → (𝑛𝐷 ∧ dom 𝑓 = 𝑛)) → (𝑓𝐵 → ∅ ∈ dom 𝑓))
2316, 22exlimi 2124 . 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 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wrex 2942  cdif 3604  c0 3948  {csn 4210  dom cdm 5143   Fn wfn 5921  ωcom 7107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-fn 5929  df-om 7108
This theorem is referenced by:  bnj906  31126
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