Theorem fnunsn 5191

Description: Extension of a function with a new ordered pair. (Contributed by NM, 28-Sep-2013.)

Hypotheses

Ref	Expression
fnunop.x	⊢ (φ → X ∈ V)
fnunop.y	⊢ (φ → Y ∈ V)
fnunop.f	⊢ (φ → F Fn D)
fnunop.g	⊢ G = (F ∪ {⟨X, Y⟩})
fnunop.e	⊢ E = (D ∪ {X})
fnunop.d	⊢ (φ → ¬ X ∈ D)

Assertion

Ref	Expression
fnunsn	⊢ (φ → G Fn E)

Proof of Theorem fnunsn

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnunop.f	. . 3 ⊢ (φ → F Fn D)
2		fnunop.x	. . . 4 ⊢ (φ → X ∈ V)
3		fnunop.y	. . . 4 ⊢ (φ → Y ∈ V)
4		opeq1 4579	. . . . . . 7 ⊢ (x = X → ⟨x, y⟩ = ⟨X, y⟩)
5	4	sneqd 3747	. . . . . 6 ⊢ (x = X → {⟨x, y⟩} = {⟨X, y⟩})
6		sneq 3745	. . . . . 6 ⊢ (x = X → {x} = {X})
7	5, 6	fneq12d 5178	. . . . 5 ⊢ (x = X → ({⟨x, y⟩} Fn {x} ↔ {⟨X, y⟩} Fn {X}))
8		opeq2 4580	. . . . . . 7 ⊢ (y = Y → ⟨X, y⟩ = ⟨X, Y⟩)
9	8	sneqd 3747	. . . . . 6 ⊢ (y = Y → {⟨X, y⟩} = {⟨X, Y⟩})
10	9	fneq1d 5176	. . . . 5 ⊢ (y = Y → ({⟨X, y⟩} Fn {X} ↔ {⟨X, Y⟩} Fn {X}))
11		vex 2863	. . . . . 6 ⊢ x ∈ V
12		vex 2863	. . . . . 6 ⊢ y ∈ V
13	11, 12	fnsn 5153	. . . . 5 ⊢ {⟨x, y⟩} Fn {x}
14	7, 10, 13	vtocl2g 2919	. . . 4 ⊢ ((X ∈ V ∧ Y ∈ V) → {⟨X, Y⟩} Fn {X})
15	2, 3, 14	syl2anc 642	. . 3 ⊢ (φ → {⟨X, Y⟩} Fn {X})
16		fnunop.d	. . . 4 ⊢ (φ → ¬ X ∈ D)
17		disjsn 3787	. . . 4 ⊢ ((D ∩ {X}) = ∅ ↔ ¬ X ∈ D)
18	16, 17	sylibr 203	. . 3 ⊢ (φ → (D ∩ {X}) = ∅)
19		fnun 5190	. . 3 ⊢ (((F Fn D ∧ {⟨X, Y⟩} Fn {X}) ∧ (D ∩ {X}) = ∅) → (F ∪ {⟨X, Y⟩}) Fn (D ∪ {X}))
20	1, 15, 18, 19	syl21anc 1181	. 2 ⊢ (φ → (F ∪ {⟨X, Y⟩}) Fn (D ∪ {X}))
21		fnunop.g	. . . 4 ⊢ G = (F ∪ {⟨X, Y⟩})
22	21	fneq1i 5179	. . 3 ⊢ (G Fn E ↔ (F ∪ {⟨X, Y⟩}) Fn E)
23		fnunop.e	. . . 4 ⊢ E = (D ∪ {X})
24	23	fneq2i 5180	. . 3 ⊢ ((F ∪ {⟨X, Y⟩}) Fn E ↔ (F ∪ {⟨X, Y⟩}) Fn (D ∪ {X}))
25	22, 24	bitri 240	. 2 ⊢ (G Fn E ↔ (F ∪ {⟨X, Y⟩}) Fn (D ∪ {X}))
26	20, 25	sylibr 203	1 ⊢ (φ → G Fn E)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ∪ cun 3208   ∩ cin 3209  ∅c0 3551  {csn 3738  ⟨cop 4562   Fn wfn 4777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791

This theorem is referenced by: (None)