Theorem bnj945 34756

Description: Technical lemma for bnj69 34993. 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 6603	. . . . . . 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 2814	. . . . 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 34755	. . . . . . . 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 6601	. . . . . 6 ⊢ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) → Fun 𝐺)
9	5	bnj931 34753	. . . . . 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 34670	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝐴 ∈ 𝑛) ↔ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛) ∧ 𝐴 ∈ 𝑛))
14		3ancomb 1098	. . . . . 6 ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛) ↔ (𝐶 ∈ V ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛))
15		3anass 1094	. . . . . 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 1088	. . 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 6861	. 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 1086   = wceq 1540   ∈ wcel 2109  Vcvv 3444   ∪ cun 3909   ⊆ wss 3911  {csn 4585  ⟨cop 4591  dom cdm 5631  suc csuc 6322  Fun wfun 6493   Fn wfn 6494  ‘cfv 6499   ∧ w-bnj17 34669

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-reg 9521

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-res 5643  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-fv 6507  df-bnj17 34670

This theorem is referenced by:  bnj966  34927  bnj967  34928  bnj1006  34943