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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 6014 | . . . 4 ⊢ ((𝐴 ∪ {〈𝐵, 𝐶〉}) ↾ {𝐷}) = ((𝐴 ↾ {𝐷}) ∪ ({〈𝐵, 𝐶〉} ↾ {𝐷})) | |
2 | nelsn 4670 | . . . . . . 7 ⊢ (𝐵 ≠ 𝐷 → ¬ 𝐵 ∈ {𝐷}) | |
3 | ressnop0 7172 | . . . . . . 7 ⊢ (¬ 𝐵 ∈ {𝐷} → ({〈𝐵, 𝐶〉} ↾ {𝐷}) = ∅) | |
4 | 2, 3 | syl 17 | . . . . . 6 ⊢ (𝐵 ≠ 𝐷 → ({〈𝐵, 𝐶〉} ↾ {𝐷}) = ∅) |
5 | 4 | uneq2d 4177 | . . . . 5 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ↾ {𝐷}) ∪ ({〈𝐵, 𝐶〉} ↾ {𝐷})) = ((𝐴 ↾ {𝐷}) ∪ ∅)) |
6 | un0 4399 | . . . . 5 ⊢ ((𝐴 ↾ {𝐷}) ∪ ∅) = (𝐴 ↾ {𝐷}) | |
7 | 5, 6 | eqtrdi 2790 | . . . 4 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ↾ {𝐷}) ∪ ({〈𝐵, 𝐶〉} ↾ {𝐷})) = (𝐴 ↾ {𝐷})) |
8 | 1, 7 | eqtrid 2786 | . . 3 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ∪ {〈𝐵, 𝐶〉}) ↾ {𝐷}) = (𝐴 ↾ {𝐷})) |
9 | 8 | fveq1d 6908 | . 2 ⊢ (𝐵 ≠ 𝐷 → (((𝐴 ∪ {〈𝐵, 𝐶〉}) ↾ {𝐷})‘𝐷) = ((𝐴 ↾ {𝐷})‘𝐷)) |
10 | fvressn 7181 | . . 3 ⊢ (𝐷 ∈ V → (((𝐴 ∪ {〈𝐵, 𝐶〉}) ↾ {𝐷})‘𝐷) = ((𝐴 ∪ {〈𝐵, 𝐶〉})‘𝐷)) | |
11 | fvprc 6898 | . . . 4 ⊢ (¬ 𝐷 ∈ V → (((𝐴 ∪ {〈𝐵, 𝐶〉}) ↾ {𝐷})‘𝐷) = ∅) | |
12 | fvprc 6898 | . . . 4 ⊢ (¬ 𝐷 ∈ V → ((𝐴 ∪ {〈𝐵, 𝐶〉})‘𝐷) = ∅) | |
13 | 11, 12 | eqtr4d 2777 | . . 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 7181 | . . 3 ⊢ (𝐷 ∈ V → ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴‘𝐷)) | |
16 | fvprc 6898 | . . . 4 ⊢ (¬ 𝐷 ∈ V → ((𝐴 ↾ {𝐷})‘𝐷) = ∅) | |
17 | fvprc 6898 | . . . 4 ⊢ (¬ 𝐷 ∈ V → (𝐴‘𝐷) = ∅) | |
18 | 16, 17 | eqtr4d 2777 | . . 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 2797 | 1 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ∪ {〈𝐵, 𝐶〉})‘𝐷) = (𝐴‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 Vcvv 3477 ∪ cun 3960 ∅c0 4338 {csn 4630 〈cop 4636 ↾ cres 5690 ‘cfv 6562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-xp 5694 df-res 5700 df-iota 6515 df-fv 6570 |
This theorem is referenced by: fvpr1g 7209 fvtp1 7214 fvtp1g 7217 f1ounsn 7291 ac6sfi 9317 cats1un 14755 ruclem6 16267 ruclem7 16268 wlkp1lem5 29709 wlkp1lem6 29710 fnchoice 44966 nnsum4primeseven 47724 nnsum4primesevenALTV 47725 |
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