Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj941 Structured version   Visualization version   GIF version

Theorem bnj941 31617
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj941.1 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
Assertion
Ref Expression
bnj941 (𝐶 ∈ V → ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝐺 Fn 𝑝))

Proof of Theorem bnj941
StepHypRef Expression
1 bnj941.1 . . . . 5 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
2 opeq2 4705 . . . . . . 7 (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → ⟨𝑛, 𝐶⟩ = ⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩)
32sneqd 4478 . . . . . 6 (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → {⟨𝑛, 𝐶⟩} = {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩})
43uneq2d 4055 . . . . 5 (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (𝑓 ∪ {⟨𝑛, 𝐶⟩}) = (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}))
51, 4syl5eq 2841 . . . 4 (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → 𝐺 = (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}))
65fneq1d 6308 . . 3 (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (𝐺 Fn 𝑝 ↔ (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) Fn 𝑝))
76imbi2d 342 . 2 (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝐺 Fn 𝑝) ↔ ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) Fn 𝑝)))
8 eqid 2793 . . 3 (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) = (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩})
9 0ex 5096 . . . 4 ∅ ∈ V
109elimel 4442 . . 3 if(𝐶 ∈ V, 𝐶, ∅) ∈ V
118, 10bnj927 31613 . 2 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) Fn 𝑝)
127, 11dedth 4431 1 (𝐶 ∈ V → ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝐺 Fn 𝑝))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1520  wcel 2079  Vcvv 3432  cun 3852  c0 4206  ifcif 4375  {csn 4466  cop 4472  suc csuc 6060   Fn wfn 6212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pr 5214  ax-reg 8892
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-br 4957  df-opab 5019  df-id 5340  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-suc 6064  df-fun 6219  df-fn 6220
This theorem is referenced by:  bnj945  31618  bnj910  31792
  Copyright terms: Public domain W3C validator