Theorem bnj945 32653

Description: Technical lemma for bnj69 32890. 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 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝐴 ∈ 𝑛) → (𝐺‘𝐴) = (𝑓‘𝐴))

Proof of Theorem bnj945

Step	Hyp	Ref	Expression
1		fndm 6520	. . . . . . 7 ⊢ (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
2	1	ad2antll 725	. . . . . 6 ⊢ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) → dom 𝑓 = 𝑛)
3	2	eleq2d 2824	. . . . 5 ⊢ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) → (𝐴 ∈ dom 𝑓 ↔ 𝐴 ∈ 𝑛))
4	3	pm5.32i 574	. . . 4 ⊢ (((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) ∧ 𝐴 ∈ dom 𝑓) ↔ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) ∧ 𝐴 ∈ 𝑛))
5		bnj945.1	. . . . . . . . 9 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
6	5	bnj941 32652	. . . . . . . 8 ⊢ (𝐶 ∈ V → ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝))
7	6	imp 406	. . . . . . 7 ⊢ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) → 𝐺 Fn 𝑝)
8	7	fnfund 6518	. . . . . 6 ⊢ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) → Fun 𝐺)
9	5	bnj931 32650	. . . . . 6 ⊢ 𝑓 ⊆ 𝐺
10	8, 9	jctir 520	. . . . 5 ⊢ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) → (Fun 𝐺 ∧ 𝑓 ⊆ 𝐺))
11	10	anim1i 614	. . . 4 ⊢ (((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) ∧ 𝐴 ∈ dom 𝑓) → ((Fun 𝐺 ∧ 𝑓 ⊆ 𝐺) ∧ 𝐴 ∈ dom 𝑓))
12	4, 11	sylbir 234	. . 3 ⊢ (((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) ∧ 𝐴 ∈ 𝑛) → ((Fun 𝐺 ∧ 𝑓 ⊆ 𝐺) ∧ 𝐴 ∈ dom 𝑓))
13		df-bnj17 32566	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝐴 ∈ 𝑛) ↔ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛) ∧ 𝐴 ∈ 𝑛))
14		3ancomb 1097	. . . . . 6 ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛) ↔ (𝐶 ∈ V ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛))
15		3anass 1093	. . . . . 6 ⊢ ((𝐶 ∈ V ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) ↔ (𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)))
16	14, 15	bitri 274	. . . . 5 ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛) ↔ (𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)))
17	16	anbi1i 623	. . . 4 ⊢ (((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛) ∧ 𝐴 ∈ 𝑛) ↔ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) ∧ 𝐴 ∈ 𝑛))
18	13, 17	bitri 274	. . 3 ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝐴 ∈ 𝑛) ↔ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) ∧ 𝐴 ∈ 𝑛))
19		df-3an 1087	. . 3 ⊢ ((Fun 𝐺 ∧ 𝑓 ⊆ 𝐺 ∧ 𝐴 ∈ dom 𝑓) ↔ ((Fun 𝐺 ∧ 𝑓 ⊆ 𝐺) ∧ 𝐴 ∈ dom 𝑓))
20	12, 18, 19	3imtr4i 291	. 2 ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝐴 ∈ 𝑛) → (Fun 𝐺 ∧ 𝑓 ⊆ 𝐺 ∧ 𝐴 ∈ dom 𝑓))
21		funssfv 6777	. 2 ⊢ ((Fun 𝐺 ∧ 𝑓 ⊆ 𝐺 ∧ 𝐴 ∈ dom 𝑓) → (𝐺‘𝐴) = (𝑓‘𝐴))
22	20, 21	syl 17	1 ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝐴 ∈ 𝑛) → (𝐺‘𝐴) = (𝑓‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ∪ cun 3881   ⊆ wss 3883  {csn 4558  ⟨cop 4564  dom cdm 5580  suc csuc 6253  Fun wfun 6412   Fn wfn 6413  ‘cfv 6418   ∧ w-bnj17 32565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-reg 9281

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-res 5592  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-fv 6426  df-bnj17 32566

This theorem is referenced by:  bnj966  32824  bnj967  32825  bnj1006  32840