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Mirrors > Home > MPE Home > Th. List > fvunsn | Structured version Visualization version GIF version |
Description: Remove an ordered pair not participating in a function value. (Contributed by NM, 1-Oct-2013.) (Revised by Mario Carneiro, 28-May-2014.) |
Ref | Expression |
---|---|
fvunsn | ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷) = (𝐴‘𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resundir 5953 | . . . 4 ⊢ ((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷}) = ((𝐴 ↾ {𝐷}) ∪ ({⟨𝐵, 𝐶⟩} ↾ {𝐷})) | |
2 | nelsn 4627 | . . . . . . 7 ⊢ (𝐵 ≠ 𝐷 → ¬ 𝐵 ∈ {𝐷}) | |
3 | ressnop0 7100 | . . . . . . 7 ⊢ (¬ 𝐵 ∈ {𝐷} → ({⟨𝐵, 𝐶⟩} ↾ {𝐷}) = ∅) | |
4 | 2, 3 | syl 17 | . . . . . 6 ⊢ (𝐵 ≠ 𝐷 → ({⟨𝐵, 𝐶⟩} ↾ {𝐷}) = ∅) |
5 | 4 | uneq2d 4124 | . . . . 5 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ↾ {𝐷}) ∪ ({⟨𝐵, 𝐶⟩} ↾ {𝐷})) = ((𝐴 ↾ {𝐷}) ∪ ∅)) |
6 | un0 4351 | . . . . 5 ⊢ ((𝐴 ↾ {𝐷}) ∪ ∅) = (𝐴 ↾ {𝐷}) | |
7 | 5, 6 | eqtrdi 2789 | . . . 4 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ↾ {𝐷}) ∪ ({⟨𝐵, 𝐶⟩} ↾ {𝐷})) = (𝐴 ↾ {𝐷})) |
8 | 1, 7 | eqtrid 2785 | . . 3 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷}) = (𝐴 ↾ {𝐷})) |
9 | 8 | fveq1d 6845 | . 2 ⊢ (𝐵 ≠ 𝐷 → (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ((𝐴 ↾ {𝐷})‘𝐷)) |
10 | fvressn 7109 | . . 3 ⊢ (𝐷 ∈ V → (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷)) | |
11 | fvprc 6835 | . . . 4 ⊢ (¬ 𝐷 ∈ V → (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ∅) | |
12 | fvprc 6835 | . . . 4 ⊢ (¬ 𝐷 ∈ V → ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷) = ∅) | |
13 | 11, 12 | eqtr4d 2776 | . . 3 ⊢ (¬ 𝐷 ∈ V → (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷)) |
14 | 10, 13 | pm2.61i 182 | . 2 ⊢ (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷) |
15 | fvressn 7109 | . . 3 ⊢ (𝐷 ∈ V → ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴‘𝐷)) | |
16 | fvprc 6835 | . . . 4 ⊢ (¬ 𝐷 ∈ V → ((𝐴 ↾ {𝐷})‘𝐷) = ∅) | |
17 | fvprc 6835 | . . . 4 ⊢ (¬ 𝐷 ∈ V → (𝐴‘𝐷) = ∅) | |
18 | 16, 17 | eqtr4d 2776 | . . 3 ⊢ (¬ 𝐷 ∈ V → ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴‘𝐷)) |
19 | 15, 18 | pm2.61i 182 | . 2 ⊢ ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴‘𝐷) |
20 | 9, 14, 19 | 3eqtr3g 2796 | 1 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷) = (𝐴‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 Vcvv 3444 ∪ cun 3909 ∅c0 4283 {csn 4587 ⟨cop 4593 ↾ cres 5636 ‘cfv 6497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-xp 5640 df-res 5646 df-iota 6449 df-fv 6505 |
This theorem is referenced by: fvpr1g 7137 fvpr2gOLD 7139 fvpr1OLD 7141 fvtp1 7145 fvtp1g 7148 ac6sfi 9234 cats1un 14615 ruclem6 16122 ruclem7 16123 wlkp1lem5 28667 wlkp1lem6 28668 fnchoice 43322 nnsum4primeseven 46078 nnsum4primesevenALTV 46079 |
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