Theorem fnunsn 5190

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 4578	. . . . . . 7 ⊢ (x = X → ⟨x, y⟩ = ⟨X, y⟩)
5	4	sneqd 3746	. . . . . 6 ⊢ (x = X → {⟨x, y⟩} = {⟨X, y⟩})
6		sneq 3744	. . . . . 6 ⊢ (x = X → {x} = {X})
7	5, 6	fneq12d 5177	. . . . 5 ⊢ (x = X → ({⟨x, y⟩} Fn {x} ↔ {⟨X, y⟩} Fn {X}))
8		opeq2 4579	. . . . . . 7 ⊢ (y = Y → ⟨X, y⟩ = ⟨X, Y⟩)
9	8	sneqd 3746	. . . . . 6 ⊢ (y = Y → {⟨X, y⟩} = {⟨X, Y⟩})
10	9	fneq1d 5175	. . . . 5 ⊢ (y = Y → ({⟨X, y⟩} Fn {X} ↔ {⟨X, Y⟩} Fn {X}))
11		vex 2862	. . . . . 6 ⊢ x ∈ V
12		vex 2862	. . . . . 6 ⊢ y ∈ V
13	11, 12	fnsn 5152	. . . . 5 ⊢ {⟨x, y⟩} Fn {x}
14	7, 10, 13	vtocl2g 2918	. . . 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 3786	. . . 4 ⊢ ((D ∩ {X}) = ∅ ↔ ¬ X ∈ D)
18	16, 17	sylibr 203	. . 3 ⊢ (φ → (D ∩ {X}) = ∅)
19		fnun 5189	. . 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 5178	. . 3 ⊢ (G Fn E ↔ (F ∪ {⟨X, Y⟩}) Fn E)
23		fnunop.e	. . . 4 ⊢ E = (D ∪ {X})
24	23	fneq2i 5179	. . 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 2859   ∪ cun 3207   ∩ cin 3208  ∅c0 3550  {csn 3737  ⟨cop 4561   Fn wfn 4776

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790

This theorem is referenced by: (None)