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Theorem bnj945 33425
Description: Technical lemma for bnj69 33662. 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 6610 . . . . . . 7 (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
21ad2antll 728 . . . . . 6 ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) → dom 𝑓 = 𝑛)
32eleq2d 2824 . . . . 5 ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) → (𝐴 ∈ dom 𝑓𝐴𝑛))
43pm5.32i 576 . . . 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 33424 . . . . . . . 8 (𝐶 ∈ V → ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝐺 Fn 𝑝))
76imp 408 . . . . . . 7 ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) → 𝐺 Fn 𝑝)
87fnfund 6608 . . . . . 6 ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) → Fun 𝐺)
95bnj931 33422 . . . . . 6 𝑓𝐺
108, 9jctir 522 . . . . 5 ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) → (Fun 𝐺𝑓𝐺))
1110anim1i 616 . . . 4 (((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) ∧ 𝐴 ∈ dom 𝑓) → ((Fun 𝐺𝑓𝐺) ∧ 𝐴 ∈ dom 𝑓))
124, 11sylbir 234 . . 3 (((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) ∧ 𝐴𝑛) → ((Fun 𝐺𝑓𝐺) ∧ 𝐴 ∈ dom 𝑓))
13 df-bnj17 33339 . . . 4 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝐴𝑛) ↔ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛) ∧ 𝐴𝑛))
14 3ancomb 1100 . . . . . 6 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛) ↔ (𝐶 ∈ V ∧ 𝑝 = suc 𝑛𝑓 Fn 𝑛))
15 3anass 1096 . . . . . 6 ((𝐶 ∈ V ∧ 𝑝 = suc 𝑛𝑓 Fn 𝑛) ↔ (𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)))
1614, 15bitri 275 . . . . 5 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛) ↔ (𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)))
1716anbi1i 625 . . . 4 (((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛) ∧ 𝐴𝑛) ↔ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) ∧ 𝐴𝑛))
1813, 17bitri 275 . . 3 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝐴𝑛) ↔ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) ∧ 𝐴𝑛))
19 df-3an 1090 . . 3 ((Fun 𝐺𝑓𝐺𝐴 ∈ dom 𝑓) ↔ ((Fun 𝐺𝑓𝐺) ∧ 𝐴 ∈ dom 𝑓))
2012, 18, 193imtr4i 292 . 2 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝐴𝑛) → (Fun 𝐺𝑓𝐺𝐴 ∈ dom 𝑓))
21 funssfv 6868 . 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 397  w3a 1088   = wceq 1542  wcel 2107  Vcvv 3448  cun 3913  wss 3915  {csn 4591  cop 4597  dom cdm 5638  suc csuc 6324  Fun wfun 6495   Fn wfn 6496  cfv 6501  w-bnj17 33338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-reg 9535
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-res 5650  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-fv 6509  df-bnj17 33339
This theorem is referenced by:  bnj966  33596  bnj967  33597  bnj1006  33612
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