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Theorem bnj941 31178
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 4540 . . . . . . 7 (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → ⟨𝑛, 𝐶⟩ = ⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩)
32sneqd 4328 . . . . . 6 (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → {⟨𝑛, 𝐶⟩} = {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩})
43uneq2d 3918 . . . . 5 (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (𝑓 ∪ {⟨𝑛, 𝐶⟩}) = (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}))
51, 4syl5eq 2817 . . . 4 (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → 𝐺 = (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}))
65fneq1d 6121 . . 3 (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (𝐺 Fn 𝑝 ↔ (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) Fn 𝑝))
76imbi2d 329 . 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 2771 . . 3 (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) = (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩})
9 0ex 4924 . . . 4 ∅ ∈ V
109elimel 4289 . . 3 if(𝐶 ∈ V, 𝐶, ∅) ∈ V
118, 10bnj927 31174 . 2 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → (𝑓 ∪ {⟨𝑛, if(𝐶 ∈ V, 𝐶, ∅)⟩}) Fn 𝑝)
127, 11dedth 4278 1 (𝐶 ∈ V → ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝐺 Fn 𝑝))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  cun 3721  c0 4063  ifcif 4225  {csn 4316  cop 4322  suc csuc 5868   Fn wfn 6026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-reg 8652
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-suc 5872  df-fun 6033  df-fn 6034
This theorem is referenced by:  bnj945  31179  bnj910  31353
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