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Theorem bnj945 30818
Description: Technical lemma for bnj69 31052. 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.)
Hypothesis
Ref Expression
bnj945.1 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
Assertion
Ref Expression
bnj945 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝐴𝑛) → (𝐺𝐴) = (𝑓𝐴))

Proof of Theorem bnj945
StepHypRef Expression
1 fndm 5978 . . . . . . 7 (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
21ad2antll 764 . . . . . 6 ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) → dom 𝑓 = 𝑛)
32eleq2d 2685 . . . . 5 ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) → (𝐴 ∈ dom 𝑓𝐴𝑛))
43pm5.32i 668 . . . 4 (((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) ∧ 𝐴 ∈ dom 𝑓) ↔ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) ∧ 𝐴𝑛))
5 bnj945.1 . . . . . . . . 9 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
65bnj941 30817 . . . . . . . 8 (𝐶 ∈ V → ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝐺 Fn 𝑝))
76imp 445 . . . . . . 7 ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) → 𝐺 Fn 𝑝)
87bnj930 30814 . . . . . 6 ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) → Fun 𝐺)
95bnj931 30815 . . . . . 6 𝑓𝐺
108, 9jctir 560 . . . . 5 ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) → (Fun 𝐺𝑓𝐺))
1110anim1i 591 . . . 4 (((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) ∧ 𝐴 ∈ dom 𝑓) → ((Fun 𝐺𝑓𝐺) ∧ 𝐴 ∈ dom 𝑓))
124, 11sylbir 225 . . 3 (((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) ∧ 𝐴𝑛) → ((Fun 𝐺𝑓𝐺) ∧ 𝐴 ∈ dom 𝑓))
13 df-bnj17 30727 . . . 4 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝐴𝑛) ↔ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛) ∧ 𝐴𝑛))
14 3ancomb 1045 . . . . . 6 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛) ↔ (𝐶 ∈ V ∧ 𝑝 = suc 𝑛𝑓 Fn 𝑛))
15 3anass 1040 . . . . . 6 ((𝐶 ∈ V ∧ 𝑝 = suc 𝑛𝑓 Fn 𝑛) ↔ (𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)))
1614, 15bitri 264 . . . . 5 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛) ↔ (𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)))
1716anbi1i 730 . . . 4 (((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛) ∧ 𝐴𝑛) ↔ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) ∧ 𝐴𝑛))
1813, 17bitri 264 . . 3 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝐴𝑛) ↔ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) ∧ 𝐴𝑛))
19 df-3an 1038 . . 3 ((Fun 𝐺𝑓𝐺𝐴 ∈ dom 𝑓) ↔ ((Fun 𝐺𝑓𝐺) ∧ 𝐴 ∈ dom 𝑓))
2012, 18, 193imtr4i 281 . 2 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝐴𝑛) → (Fun 𝐺𝑓𝐺𝐴 ∈ dom 𝑓))
21 funssfv 6196 . 2 ((Fun 𝐺𝑓𝐺𝐴 ∈ dom 𝑓) → (𝐺𝐴) = (𝑓𝐴))
2220, 21syl 17 1 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝐴𝑛) → (𝐺𝐴) = (𝑓𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1481  wcel 1988  Vcvv 3195  cun 3565  wss 3567  {csn 4168  cop 4174  dom cdm 5104  suc csuc 5713  Fun wfun 5870   Fn wfn 5871  cfv 5876  w-bnj17 30726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-reg 8482
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-res 5116  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-fv 5884  df-bnj17 30727
This theorem is referenced by:  bnj966  30988  bnj967  30989  bnj1006  31003
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