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Theorem bnj927 29899
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj927.1 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
bnj927.2 𝐶 ∈ V
Assertion
Ref Expression
bnj927 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)

Proof of Theorem bnj927
StepHypRef Expression
1 simpr 475 . . . 4 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝑓 Fn 𝑛)
2 vex 3175 . . . . . 6 𝑛 ∈ V
3 bnj927.2 . . . . . 6 𝐶 ∈ V
42, 3fnsn 5846 . . . . 5 {⟨𝑛, 𝐶⟩} Fn {𝑛}
54a1i 11 . . . 4 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → {⟨𝑛, 𝐶⟩} Fn {𝑛})
6 bnj521 29865 . . . . 5 (𝑛 ∩ {𝑛}) = ∅
76a1i 11 . . . 4 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → (𝑛 ∩ {𝑛}) = ∅)
8 fnun 5897 . . . 4 (((𝑓 Fn 𝑛 ∧ {⟨𝑛, 𝐶⟩} Fn {𝑛}) ∧ (𝑛 ∩ {𝑛}) = ∅) → (𝑓 ∪ {⟨𝑛, 𝐶⟩}) Fn (𝑛 ∪ {𝑛}))
91, 5, 7, 8syl21anc 1316 . . 3 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → (𝑓 ∪ {⟨𝑛, 𝐶⟩}) Fn (𝑛 ∪ {𝑛}))
10 bnj927.1 . . . 4 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
1110fneq1i 5885 . . 3 (𝐺 Fn (𝑛 ∪ {𝑛}) ↔ (𝑓 ∪ {⟨𝑛, 𝐶⟩}) Fn (𝑛 ∪ {𝑛}))
129, 11sylibr 222 . 2 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝐺 Fn (𝑛 ∪ {𝑛}))
13 df-suc 5632 . . . . . 6 suc 𝑛 = (𝑛 ∪ {𝑛})
1413eqeq2i 2621 . . . . 5 (𝑝 = suc 𝑛𝑝 = (𝑛 ∪ {𝑛}))
1514biimpi 204 . . . 4 (𝑝 = suc 𝑛𝑝 = (𝑛 ∪ {𝑛}))
1615adantr 479 . . 3 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝑝 = (𝑛 ∪ {𝑛}))
1716fneq2d 5882 . 2 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → (𝐺 Fn 𝑝𝐺 Fn (𝑛 ∪ {𝑛})))
1812, 17mpbird 245 1 ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  Vcvv 3172  cun 3537  cin 3538  c0 3873  {csn 4124  cop 4130  suc csuc 5628   Fn wfn 5785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828  ax-reg 8357
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-suc 5632  df-fun 5792  df-fn 5793
This theorem is referenced by:  bnj941  29903  bnj929  30066
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