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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 5851 | . . . 4 ⊢ ((𝐴 ∪ {〈𝐵, 𝐶〉}) ↾ {𝐷}) = ((𝐴 ↾ {𝐷}) ∪ ({〈𝐵, 𝐶〉} ↾ {𝐷})) | |
2 | nelsn 4567 | . . . . . . 7 ⊢ (𝐵 ≠ 𝐷 → ¬ 𝐵 ∈ {𝐷}) | |
3 | ressnop0 6946 | . . . . . . 7 ⊢ (¬ 𝐵 ∈ {𝐷} → ({〈𝐵, 𝐶〉} ↾ {𝐷}) = ∅) | |
4 | 2, 3 | syl 17 | . . . . . 6 ⊢ (𝐵 ≠ 𝐷 → ({〈𝐵, 𝐶〉} ↾ {𝐷}) = ∅) |
5 | 4 | uneq2d 4063 | . . . . 5 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ↾ {𝐷}) ∪ ({〈𝐵, 𝐶〉} ↾ {𝐷})) = ((𝐴 ↾ {𝐷}) ∪ ∅)) |
6 | un0 4291 | . . . . 5 ⊢ ((𝐴 ↾ {𝐷}) ∪ ∅) = (𝐴 ↾ {𝐷}) | |
7 | 5, 6 | eqtrdi 2787 | . . . 4 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ↾ {𝐷}) ∪ ({〈𝐵, 𝐶〉} ↾ {𝐷})) = (𝐴 ↾ {𝐷})) |
8 | 1, 7 | syl5eq 2783 | . . 3 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ∪ {〈𝐵, 𝐶〉}) ↾ {𝐷}) = (𝐴 ↾ {𝐷})) |
9 | 8 | fveq1d 6697 | . 2 ⊢ (𝐵 ≠ 𝐷 → (((𝐴 ∪ {〈𝐵, 𝐶〉}) ↾ {𝐷})‘𝐷) = ((𝐴 ↾ {𝐷})‘𝐷)) |
10 | fvressn 6955 | . . 3 ⊢ (𝐷 ∈ V → (((𝐴 ∪ {〈𝐵, 𝐶〉}) ↾ {𝐷})‘𝐷) = ((𝐴 ∪ {〈𝐵, 𝐶〉})‘𝐷)) | |
11 | fvprc 6687 | . . . 4 ⊢ (¬ 𝐷 ∈ V → (((𝐴 ∪ {〈𝐵, 𝐶〉}) ↾ {𝐷})‘𝐷) = ∅) | |
12 | fvprc 6687 | . . . 4 ⊢ (¬ 𝐷 ∈ V → ((𝐴 ∪ {〈𝐵, 𝐶〉})‘𝐷) = ∅) | |
13 | 11, 12 | eqtr4d 2774 | . . 3 ⊢ (¬ 𝐷 ∈ V → (((𝐴 ∪ {〈𝐵, 𝐶〉}) ↾ {𝐷})‘𝐷) = ((𝐴 ∪ {〈𝐵, 𝐶〉})‘𝐷)) |
14 | 10, 13 | pm2.61i 185 | . 2 ⊢ (((𝐴 ∪ {〈𝐵, 𝐶〉}) ↾ {𝐷})‘𝐷) = ((𝐴 ∪ {〈𝐵, 𝐶〉})‘𝐷) |
15 | fvressn 6955 | . . 3 ⊢ (𝐷 ∈ V → ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴‘𝐷)) | |
16 | fvprc 6687 | . . . 4 ⊢ (¬ 𝐷 ∈ V → ((𝐴 ↾ {𝐷})‘𝐷) = ∅) | |
17 | fvprc 6687 | . . . 4 ⊢ (¬ 𝐷 ∈ V → (𝐴‘𝐷) = ∅) | |
18 | 16, 17 | eqtr4d 2774 | . . 3 ⊢ (¬ 𝐷 ∈ V → ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴‘𝐷)) |
19 | 15, 18 | pm2.61i 185 | . 2 ⊢ ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴‘𝐷) |
20 | 9, 14, 19 | 3eqtr3g 2794 | 1 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ∪ {〈𝐵, 𝐶〉})‘𝐷) = (𝐴‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 Vcvv 3398 ∪ cun 3851 ∅c0 4223 {csn 4527 〈cop 4533 ↾ cres 5538 ‘cfv 6358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-xp 5542 df-res 5548 df-iota 6316 df-fv 6366 |
This theorem is referenced by: fvpr1 6983 fvpr1g 6985 fvpr2g 6986 fvtp1 6988 fvtp1g 6991 ac6sfi 8893 cats1un 14251 ruclem6 15759 ruclem7 15760 wlkp1lem5 27719 wlkp1lem6 27720 fnchoice 42186 nnsum4primeseven 44868 nnsum4primesevenALTV 44869 |
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