Theorem bnj945 34787

Description: Technical lemma for bnj69 35024. 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 6671	. . . . . . 7 ⊢ (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
2	1	ad2antll 729	. . . . . 6 ⊢ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) → dom 𝑓 = 𝑛)
3	2	eleq2d 2827	. . . . 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 34786	. . . . . . . 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 6669	. . . . . 6 ⊢ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) → Fun 𝐺)
9	5	bnj931 34784	. . . . . 6 ⊢ 𝑓 ⊆ 𝐺
10	8, 9	jctir 520	. . . . 5 ⊢ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) → (Fun 𝐺 ∧ 𝑓 ⊆ 𝐺))
11	10	anim1i 615	. . . 4 ⊢ (((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) ∧ 𝐴 ∈ dom 𝑓) → ((Fun 𝐺 ∧ 𝑓 ⊆ 𝐺) ∧ 𝐴 ∈ dom 𝑓))
12	4, 11	sylbir 235	. . 3 ⊢ (((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) ∧ 𝐴 ∈ 𝑛) → ((Fun 𝐺 ∧ 𝑓 ⊆ 𝐺) ∧ 𝐴 ∈ dom 𝑓))
13		df-bnj17 34701	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝐴 ∈ 𝑛) ↔ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛) ∧ 𝐴 ∈ 𝑛))
14		3ancomb 1099	. . . . . 6 ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛) ↔ (𝐶 ∈ V ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛))
15		3anass 1095	. . . . . 6 ⊢ ((𝐶 ∈ V ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) ↔ (𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)))
16	14, 15	bitri 275	. . . . 5 ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛) ↔ (𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)))
17	16	anbi1i 624	. . . 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 275	. . 3 ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝐴 ∈ 𝑛) ↔ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) ∧ 𝐴 ∈ 𝑛))
19		df-3an 1089	. . 3 ⊢ ((Fun 𝐺 ∧ 𝑓 ⊆ 𝐺 ∧ 𝐴 ∈ dom 𝑓) ↔ ((Fun 𝐺 ∧ 𝑓 ⊆ 𝐺) ∧ 𝐴 ∈ dom 𝑓))
20	12, 18, 19	3imtr4i 292	. 2 ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝐴 ∈ 𝑛) → (Fun 𝐺 ∧ 𝑓 ⊆ 𝐺 ∧ 𝐴 ∈ dom 𝑓))
21		funssfv 6927	. 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 1087   = wceq 1540   ∈ wcel 2108  Vcvv 3480   ∪ cun 3949   ⊆ wss 3951  {csn 4626  ⟨cop 4632  dom cdm 5685  suc csuc 6386  Fun wfun 6555   Fn wfn 6556  ‘cfv 6561   ∧ w-bnj17 34700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-reg 9632

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-bnj17 34701

This theorem is referenced by:  bnj966  34958  bnj967  34959  bnj1006  34974