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Theorem bnj945 29901
Description: Technical lemma for bnj69 30135. 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 5887 . . . . . . 7 (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
21ad2antll 760 . . . . . 6 ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) → dom 𝑓 = 𝑛)
32eleq2d 2669 . . . . 5 ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) → (𝐴 ∈ dom 𝑓𝐴𝑛))
43pm5.32i 666 . . . 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 29900 . . . . . . . 8 (𝐶 ∈ V → ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝐺 Fn 𝑝))
76imp 443 . . . . . . 7 ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) → 𝐺 Fn 𝑝)
87bnj930 29897 . . . . . 6 ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) → Fun 𝐺)
95bnj931 29898 . . . . . 6 𝑓𝐺
108, 9jctir 558 . . . . 5 ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) → (Fun 𝐺𝑓𝐺))
1110anim1i 589 . . . 4 (((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) ∧ 𝐴 ∈ dom 𝑓) → ((Fun 𝐺𝑓𝐺) ∧ 𝐴 ∈ dom 𝑓))
124, 11sylbir 223 . . 3 (((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) ∧ 𝐴𝑛) → ((Fun 𝐺𝑓𝐺) ∧ 𝐴 ∈ dom 𝑓))
13 df-bnj17 29809 . . . 4 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝐴𝑛) ↔ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛) ∧ 𝐴𝑛))
14 3ancomb 1039 . . . . . 6 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛) ↔ (𝐶 ∈ V ∧ 𝑝 = suc 𝑛𝑓 Fn 𝑛))
15 3anass 1034 . . . . . 6 ((𝐶 ∈ V ∧ 𝑝 = suc 𝑛𝑓 Fn 𝑛) ↔ (𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)))
1614, 15bitri 262 . . . . 5 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛) ↔ (𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)))
1716anbi1i 726 . . . 4 (((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛) ∧ 𝐴𝑛) ↔ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) ∧ 𝐴𝑛))
1813, 17bitri 262 . . 3 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝐴𝑛) ↔ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) ∧ 𝐴𝑛))
19 df-3an 1032 . . 3 ((Fun 𝐺𝑓𝐺𝐴 ∈ dom 𝑓) ↔ ((Fun 𝐺𝑓𝐺) ∧ 𝐴 ∈ dom 𝑓))
2012, 18, 193imtr4i 279 . 2 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝐴𝑛) → (Fun 𝐺𝑓𝐺𝐴 ∈ dom 𝑓))
21 funssfv 6101 . 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 382  w3a 1030   = wceq 1474  wcel 1976  Vcvv 3169  cun 3534  wss 3536  {csn 4121  cop 4127  dom cdm 5025  suc csuc 5625  Fun wfun 5781   Fn wfn 5782  cfv 5787  w-bnj17 29808
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 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pr 4825  ax-reg 8354
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 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-br 4575  df-opab 4635  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-res 5037  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-fv 5795  df-bnj17 29809
This theorem is referenced by:  bnj966  30071  bnj967  30072  bnj1006  30086
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