Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj900 Structured version   Visualization version   GIF version

Theorem bnj900 34469
Description: Technical lemma for bnj69 34550. 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 34379 . . . . 5 (𝑓𝐵 → ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓))
3 simp1 1133 . . . . . 6 ((𝑓 Fn 𝑛𝜑𝜓) → 𝑓 Fn 𝑛)
43reximi 3078 . . . . 5 (∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓) → ∃𝑛𝐷 𝑓 Fn 𝑛)
5 fndm 6645 . . . . . 6 (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
65reximi 3078 . . . . 5 (∃𝑛𝐷 𝑓 Fn 𝑛 → ∃𝑛𝐷 dom 𝑓 = 𝑛)
72, 4, 63syl 18 . . . 4 (𝑓𝐵 → ∃𝑛𝐷 dom 𝑓 = 𝑛)
87bnj1196 34334 . . 3 (𝑓𝐵 → ∃𝑛(𝑛𝐷 ∧ dom 𝑓 = 𝑛))
9 nfre1 3276 . . . . . . 7 𝑛𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)
109nfab 2903 . . . . . 6 𝑛{𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
111, 10nfcxfr 2895 . . . . 5 𝑛𝐵
1211nfcri 2884 . . . 4 𝑛 𝑓𝐵
131219.37 2217 . . 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 1909 . . . 4 𝑛∅ ∈ dom 𝑓
1612, 15nfim 1891 . . 3 𝑛(𝑓𝐵 → ∅ ∈ dom 𝑓)
17 bnj900.3 . . . . . 6 𝐷 = (ω ∖ {∅})
1817bnj529 34281 . . . . 5 (𝑛𝐷 → ∅ ∈ 𝑛)
19 eleq2 2816 . . . . . 6 (dom 𝑓 = 𝑛 → (∅ ∈ dom 𝑓 ↔ ∅ ∈ 𝑛))
2019biimparc 479 . . . . 5 ((∅ ∈ 𝑛 ∧ dom 𝑓 = 𝑛) → ∅ ∈ dom 𝑓)
2118, 20sylan 579 . . . 4 ((𝑛𝐷 ∧ dom 𝑓 = 𝑛) → ∅ ∈ dom 𝑓)
2221imim2i 16 . . 3 ((𝑓𝐵 → (𝑛𝐷 ∧ dom 𝑓 = 𝑛)) → (𝑓𝐵 → ∅ ∈ dom 𝑓))
2316, 22exlimi 2202 . 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 395  w3a 1084   = wceq 1533  wex 1773  wcel 2098  {cab 2703  wrex 3064  cdif 3940  c0 4317  {csn 4623  dom cdm 5669   Fn wfn 6531  ωcom 7851
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-tr 5259  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-ord 6360  df-on 6361  df-fn 6539  df-om 7852
This theorem is referenced by:  bnj906  34470
  Copyright terms: Public domain W3C validator