Theorem bnj945 30098

Description: Technical lemma for bnj69 30332. 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 5904	. . . . . . 7 ⊢ (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
2	1	ad2antll 761	. . . . . 6 ⊢ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) → dom 𝑓 = 𝑛)
3	2	eleq2d 2673	. . . . 5 ⊢ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) → (𝐴 ∈ dom 𝑓 ↔ 𝐴 ∈ 𝑛))
4	3	pm5.32i 667	. . . 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 30097	. . . . . . . 8 ⊢ (𝐶 ∈ V → ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝))
7	6	imp 444	. . . . . . 7 ⊢ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) → 𝐺 Fn 𝑝)
8	7	bnj930 30094	. . . . . 6 ⊢ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) → Fun 𝐺)
9	5	bnj931 30095	. . . . . 6 ⊢ 𝑓 ⊆ 𝐺
10	8, 9	jctir 559	. . . . 5 ⊢ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) → (Fun 𝐺 ∧ 𝑓 ⊆ 𝐺))
11	10	anim1i 590	. . . 4 ⊢ (((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) ∧ 𝐴 ∈ dom 𝑓) → ((Fun 𝐺 ∧ 𝑓 ⊆ 𝐺) ∧ 𝐴 ∈ dom 𝑓))
12	4, 11	sylbir 224	. . 3 ⊢ (((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) ∧ 𝐴 ∈ 𝑛) → ((Fun 𝐺 ∧ 𝑓 ⊆ 𝐺) ∧ 𝐴 ∈ dom 𝑓))
13		df-bnj17 30006	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝐴 ∈ 𝑛) ↔ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛) ∧ 𝐴 ∈ 𝑛))
14		3ancomb 1040	. . . . . 6 ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛) ↔ (𝐶 ∈ V ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛))
15		3anass 1035	. . . . . 6 ⊢ ((𝐶 ∈ V ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) ↔ (𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)))
16	14, 15	bitri 263	. . . . 5 ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛) ↔ (𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)))
17	16	anbi1i 727	. . . 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 263	. . 3 ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝐴 ∈ 𝑛) ↔ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) ∧ 𝐴 ∈ 𝑛))
19		df-3an 1033	. . 3 ⊢ ((Fun 𝐺 ∧ 𝑓 ⊆ 𝐺 ∧ 𝐴 ∈ dom 𝑓) ↔ ((Fun 𝐺 ∧ 𝑓 ⊆ 𝐺) ∧ 𝐴 ∈ dom 𝑓))
20	12, 18, 19	3imtr4i 280	. 2 ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝐴 ∈ 𝑛) → (Fun 𝐺 ∧ 𝑓 ⊆ 𝐺 ∧ 𝐴 ∈ dom 𝑓))
21		funssfv 6119	. 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 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∪ cun 3538   ⊆ wss 3540  {csn 4125  ⟨cop 4131  dom cdm 5038  suc csuc 5642  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804   ∧ w-bnj17 30005

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-reg 8380

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-bnj17 30006

This theorem is referenced by:  bnj966  30268  bnj967  30269  bnj1006  30283