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Mirrors > Home > ILE Home > Th. List > fvunsng | GIF version |
Description: Remove an ordered pair not participating in a function value. (Contributed by Jim Kingdon, 7-Jan-2019.) |
Ref | Expression |
---|---|
fvunsng | ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → ((𝐴 ∪ {〈𝐵, 𝐶〉})‘𝐷) = (𝐴‘𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snidg 3549 | . . . 4 ⊢ (𝐷 ∈ 𝑉 → 𝐷 ∈ {𝐷}) | |
2 | fvres 5438 | . . . 4 ⊢ (𝐷 ∈ {𝐷} → (((𝐴 ∪ {〈𝐵, 𝐶〉}) ↾ {𝐷})‘𝐷) = ((𝐴 ∪ {〈𝐵, 𝐶〉})‘𝐷)) | |
3 | 1, 2 | syl 14 | . . 3 ⊢ (𝐷 ∈ 𝑉 → (((𝐴 ∪ {〈𝐵, 𝐶〉}) ↾ {𝐷})‘𝐷) = ((𝐴 ∪ {〈𝐵, 𝐶〉})‘𝐷)) |
4 | resundir 4828 | . . . . 5 ⊢ ((𝐴 ∪ {〈𝐵, 𝐶〉}) ↾ {𝐷}) = ((𝐴 ↾ {𝐷}) ∪ ({〈𝐵, 𝐶〉} ↾ {𝐷})) | |
5 | elsni 3540 | . . . . . . . . 9 ⊢ (𝐵 ∈ {𝐷} → 𝐵 = 𝐷) | |
6 | 5 | necon3ai 2355 | . . . . . . . 8 ⊢ (𝐵 ≠ 𝐷 → ¬ 𝐵 ∈ {𝐷}) |
7 | ressnop0 5594 | . . . . . . . 8 ⊢ (¬ 𝐵 ∈ {𝐷} → ({〈𝐵, 𝐶〉} ↾ {𝐷}) = ∅) | |
8 | 6, 7 | syl 14 | . . . . . . 7 ⊢ (𝐵 ≠ 𝐷 → ({〈𝐵, 𝐶〉} ↾ {𝐷}) = ∅) |
9 | 8 | uneq2d 3225 | . . . . . 6 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ↾ {𝐷}) ∪ ({〈𝐵, 𝐶〉} ↾ {𝐷})) = ((𝐴 ↾ {𝐷}) ∪ ∅)) |
10 | un0 3391 | . . . . . 6 ⊢ ((𝐴 ↾ {𝐷}) ∪ ∅) = (𝐴 ↾ {𝐷}) | |
11 | 9, 10 | syl6eq 2186 | . . . . 5 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ↾ {𝐷}) ∪ ({〈𝐵, 𝐶〉} ↾ {𝐷})) = (𝐴 ↾ {𝐷})) |
12 | 4, 11 | syl5eq 2182 | . . . 4 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ∪ {〈𝐵, 𝐶〉}) ↾ {𝐷}) = (𝐴 ↾ {𝐷})) |
13 | 12 | fveq1d 5416 | . . 3 ⊢ (𝐵 ≠ 𝐷 → (((𝐴 ∪ {〈𝐵, 𝐶〉}) ↾ {𝐷})‘𝐷) = ((𝐴 ↾ {𝐷})‘𝐷)) |
14 | 3, 13 | sylan9req 2191 | . 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → ((𝐴 ∪ {〈𝐵, 𝐶〉})‘𝐷) = ((𝐴 ↾ {𝐷})‘𝐷)) |
15 | fvres 5438 | . . . 4 ⊢ (𝐷 ∈ {𝐷} → ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴‘𝐷)) | |
16 | 1, 15 | syl 14 | . . 3 ⊢ (𝐷 ∈ 𝑉 → ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴‘𝐷)) |
17 | 16 | adantr 274 | . 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴‘𝐷)) |
18 | 14, 17 | eqtrd 2170 | 1 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → ((𝐴 ∪ {〈𝐵, 𝐶〉})‘𝐷) = (𝐴‘𝐷)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ≠ wne 2306 ∪ cun 3064 ∅c0 3358 {csn 3522 〈cop 3525 ↾ cres 4536 ‘cfv 5118 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-res 4546 df-iota 5083 df-fv 5126 |
This theorem is referenced by: fvpr1 5617 fvpr1g 5619 fvpr2g 5620 fvtp1g 5621 tfrlemisucaccv 6215 tfr1onlemsucaccv 6231 tfrcllemsucaccv 6244 ac6sfi 6785 0tonninf 10205 1tonninf 10206 hashennn 10519 zfz1isolemiso 10575 |
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