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Theorem bnj945 32170
 Description: Technical lemma for bnj69 32407. 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 6426 . . . . . . 7 (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
21ad2antll 728 . . . . . 6 ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) → dom 𝑓 = 𝑛)
32eleq2d 2875 . . . . 5 ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) → (𝐴 ∈ dom 𝑓𝐴𝑛))
43pm5.32i 578 . . . 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 32169 . . . . . . . 8 (𝐶 ∈ V → ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝐺 Fn 𝑝))
76imp 410 . . . . . . 7 ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) → 𝐺 Fn 𝑝)
87bnj930 32166 . . . . . 6 ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) → Fun 𝐺)
95bnj931 32167 . . . . . 6 𝑓𝐺
108, 9jctir 524 . . . . 5 ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) → (Fun 𝐺𝑓𝐺))
1110anim1i 617 . . . 4 (((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) ∧ 𝐴 ∈ dom 𝑓) → ((Fun 𝐺𝑓𝐺) ∧ 𝐴 ∈ dom 𝑓))
124, 11sylbir 238 . . 3 (((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) ∧ 𝐴𝑛) → ((Fun 𝐺𝑓𝐺) ∧ 𝐴 ∈ dom 𝑓))
13 df-bnj17 32082 . . . 4 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝐴𝑛) ↔ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛) ∧ 𝐴𝑛))
14 3ancomb 1096 . . . . . 6 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛) ↔ (𝐶 ∈ V ∧ 𝑝 = suc 𝑛𝑓 Fn 𝑛))
15 3anass 1092 . . . . . 6 ((𝐶 ∈ V ∧ 𝑝 = suc 𝑛𝑓 Fn 𝑛) ↔ (𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)))
1614, 15bitri 278 . . . . 5 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛) ↔ (𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)))
1716anbi1i 626 . . . 4 (((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛) ∧ 𝐴𝑛) ↔ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) ∧ 𝐴𝑛))
1813, 17bitri 278 . . 3 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝐴𝑛) ↔ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) ∧ 𝐴𝑛))
19 df-3an 1086 . . 3 ((Fun 𝐺𝑓𝐺𝐴 ∈ dom 𝑓) ↔ ((Fun 𝐺𝑓𝐺) ∧ 𝐴 ∈ dom 𝑓))
2012, 18, 193imtr4i 295 . 2 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝐴𝑛) → (Fun 𝐺𝑓𝐺𝐴 ∈ dom 𝑓))
21 funssfv 6667 . 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 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∪ cun 3879   ⊆ wss 3881  {csn 4525  ⟨cop 4531  dom cdm 5520  suc csuc 6162  Fun wfun 6319   Fn wfn 6320  ‘cfv 6325   ∧ w-bnj17 32081 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296  ax-reg 9043 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-res 5532  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-fv 6333  df-bnj17 32082 This theorem is referenced by:  bnj966  32341  bnj967  32342  bnj1006  32357
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