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Theorem bnj900 30742
Description: Technical lemma for bnj69 30821. 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 30653 . . . . 5 (𝑓𝐵 → ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓))
3 simp1 1059 . . . . . 6 ((𝑓 Fn 𝑛𝜑𝜓) → 𝑓 Fn 𝑛)
43reximi 3006 . . . . 5 (∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) → ∃𝑛𝐷 𝑓 Fn 𝑛)
5 fndm 5953 . . . . . 6 (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
65reximi 3006 . . . . 5 (∃𝑛𝐷 𝑓 Fn 𝑛 → ∃𝑛𝐷 dom 𝑓 = 𝑛)
72, 4, 63syl 18 . . . 4 (𝑓𝐵 → ∃𝑛𝐷 dom 𝑓 = 𝑛)
87bnj1196 30608 . . 3 (𝑓𝐵 → ∃𝑛(𝑛𝐷 ∧ dom 𝑓 = 𝑛))
9 nfre1 3000 . . . . . . 7 𝑛𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)
109nfab 2765 . . . . . 6 𝑛{𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
111, 10nfcxfr 2759 . . . . 5 𝑛𝐵
1211nfcri 2755 . . . 4 𝑛 𝑓𝐵
131219.37 2098 . . 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 1840 . . . 4 𝑛∅ ∈ dom 𝑓
1612, 15nfim 1822 . . 3 𝑛(𝑓𝐵 → ∅ ∈ dom 𝑓)
17 bnj900.3 . . . . . 6 𝐷 = (ω ∖ {∅})
1817bnj529 30554 . . . . 5 (𝑛𝐷 → ∅ ∈ 𝑛)
19 eleq2 2687 . . . . . 6 (dom 𝑓 = 𝑛 → (∅ ∈ dom 𝑓 ↔ ∅ ∈ 𝑛))
2019biimparc 504 . . . . 5 ((∅ ∈ 𝑛 ∧ dom 𝑓 = 𝑛) → ∅ ∈ dom 𝑓)
2118, 20sylan 488 . . . 4 ((𝑛𝐷 ∧ dom 𝑓 = 𝑛) → ∅ ∈ dom 𝑓)
2221imim2i 16 . . 3 ((𝑓𝐵 → (𝑛𝐷 ∧ dom 𝑓 = 𝑛)) → (𝑓𝐵 → ∅ ∈ dom 𝑓))
2316, 22exlimi 2084 . 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 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wrex 2908  cdif 3556  c0 3896  {csn 4153  dom cdm 5079   Fn wfn 5847  ωcom 7019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-fn 5855  df-om 7020
This theorem is referenced by:  bnj906  30743
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