Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj945 Structured version   Visualization version   GIF version

Theorem bnj945 31224
Description: Technical lemma for bnj69 31458. 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 6168 . . . . . . 7 (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
21ad2antll 720 . . . . . 6 ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) → dom 𝑓 = 𝑛)
32eleq2d 2830 . . . . 5 ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) → (𝐴 ∈ dom 𝑓𝐴𝑛))
43pm5.32i 570 . . . 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 31223 . . . . . . . 8 (𝐶 ∈ V → ((𝑝 = suc 𝑛𝑓 Fn 𝑛) → 𝐺 Fn 𝑝))
76imp 395 . . . . . . 7 ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) → 𝐺 Fn 𝑝)
87bnj930 31220 . . . . . 6 ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) → Fun 𝐺)
95bnj931 31221 . . . . . 6 𝑓𝐺
108, 9jctir 516 . . . . 5 ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) → (Fun 𝐺𝑓𝐺))
1110anim1i 608 . . . 4 (((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) ∧ 𝐴 ∈ dom 𝑓) → ((Fun 𝐺𝑓𝐺) ∧ 𝐴 ∈ dom 𝑓))
124, 11sylbir 226 . . 3 (((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) ∧ 𝐴𝑛) → ((Fun 𝐺𝑓𝐺) ∧ 𝐴 ∈ dom 𝑓))
13 df-bnj17 31136 . . . 4 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝐴𝑛) ↔ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛) ∧ 𝐴𝑛))
14 3ancomb 1121 . . . . . 6 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛) ↔ (𝐶 ∈ V ∧ 𝑝 = suc 𝑛𝑓 Fn 𝑛))
15 3anass 1116 . . . . . 6 ((𝐶 ∈ V ∧ 𝑝 = suc 𝑛𝑓 Fn 𝑛) ↔ (𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)))
1614, 15bitri 266 . . . . 5 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛) ↔ (𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)))
1716anbi1i 617 . . . 4 (((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛) ∧ 𝐴𝑛) ↔ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) ∧ 𝐴𝑛))
1813, 17bitri 266 . . 3 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝐴𝑛) ↔ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛𝑓 Fn 𝑛)) ∧ 𝐴𝑛))
19 df-3an 1109 . . 3 ((Fun 𝐺𝑓𝐺𝐴 ∈ dom 𝑓) ↔ ((Fun 𝐺𝑓𝐺) ∧ 𝐴 ∈ dom 𝑓))
2012, 18, 193imtr4i 283 . 2 ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛𝑝 = suc 𝑛𝐴𝑛) → (Fun 𝐺𝑓𝐺𝐴 ∈ dom 𝑓))
21 funssfv 6396 . 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 1107   = wceq 1652  wcel 2155  Vcvv 3350  cun 3730  wss 3732  {csn 4334  cop 4340  dom cdm 5277  suc csuc 5910  Fun wfun 6062   Fn wfn 6063  cfv 6068  w-bnj17 31135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062  ax-reg 8704
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-res 5289  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-fv 6076  df-bnj17 31136
This theorem is referenced by:  bnj966  31394  bnj967  31395  bnj1006  31409
  Copyright terms: Public domain W3C validator