Theorem fvunsn 7177

Description: Remove an ordered pair not participating in a function value. (Contributed by NM, 1-Oct-2013.) (Revised by Mario Carneiro, 28-May-2014.)

Assertion

Ref	Expression
fvunsn	⊢ (𝐵 ≠ 𝐷 → ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷) = (𝐴‘𝐷))

Proof of Theorem fvunsn

Step	Hyp	Ref	Expression
1		resundir 5993	. . . 4 ⊢ ((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷}) = ((𝐴 ↾ {𝐷}) ∪ ({⟨𝐵, 𝐶⟩} ↾ {𝐷}))
2		nelsn 4632	. . . . . . 7 ⊢ (𝐵 ≠ 𝐷 → ¬ 𝐵 ∈ {𝐷})
3		ressnop0 7150	. . . . . . 7 ⊢ (¬ 𝐵 ∈ {𝐷} → ({⟨𝐵, 𝐶⟩} ↾ {𝐷}) = ∅)
4	2, 3	syl 18	. . . . . 6 ⊢ (𝐵 ≠ 𝐷 → ({⟨𝐵, 𝐶⟩} ↾ {𝐷}) = ∅)
5	4	uneq2d 4122	. . . . 5 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ↾ {𝐷}) ∪ ({⟨𝐵, 𝐶⟩} ↾ {𝐷})) = ((𝐴 ↾ {𝐷}) ∪ ∅))
6		un0 4351	. . . . 5 ⊢ ((𝐴 ↾ {𝐷}) ∪ ∅) = (𝐴 ↾ {𝐷})
7	5, 6	eqtrdi 2814	. . . 4 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ↾ {𝐷}) ∪ ({⟨𝐵, 𝐶⟩} ↾ {𝐷})) = (𝐴 ↾ {𝐷}))
8	1, 7	eqtrid 2810	. . 3 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷}) = (𝐴 ↾ {𝐷}))
9	8	fveq1d 6883	. 2 ⊢ (𝐵 ≠ 𝐷 → (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ((𝐴 ↾ {𝐷})‘𝐷))
10		fvressn 7159	. . 3 ⊢ (𝐷 ∈ V → (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷))
11		fvprc 6873	. . . 4 ⊢ (¬ 𝐷 ∈ V → (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ∅)
12		fvprc 6873	. . . 4 ⊢ (¬ 𝐷 ∈ V → ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷) = ∅)
13	11, 12	eqtr4d 2801	. . 3 ⊢ (¬ 𝐷 ∈ V → (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷))
14	10, 13	pm2.61i 184	. 2 ⊢ (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷)
15		fvressn 7159	. . 3 ⊢ (𝐷 ∈ V → ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴‘𝐷))
16		fvprc 6873	. . . 4 ⊢ (¬ 𝐷 ∈ V → ((𝐴 ↾ {𝐷})‘𝐷) = ∅)
17		fvprc 6873	. . . 4 ⊢ (¬ 𝐷 ∈ V → (𝐴‘𝐷) = ∅)
18	16, 17	eqtr4d 2801	. . 3 ⊢ (¬ 𝐷 ∈ V → ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴‘𝐷))
19	15, 18	pm2.61i 184	. 2 ⊢ ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴‘𝐷)
20	9, 14, 19	3eqtr3g 2821	1 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷) = (𝐴‘𝐷))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  Vcvv 3455   ∪ cun 3903  ∅c0 4286  {csn 4589  ⟨cop 4595   ↾ cres 5663  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-xp 5667  df-res 5673  df-iota 6492  df-fv 6544

This theorem is referenced by:  fvpr1g  7188  fvtp1  7193  fvtp1g  7196  f1ounsn  7270  ac6sfi  9240  cats1un  14754  ruclem6  16286  ruclem7  16287  wlkp1lem5  30025  wlkp1lem6  30026  fnchoice  45749  nnsum4primeseven  48565  nnsum4primesevenALTV  48566