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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 3561 | . . . 4 ⊢ (𝐷 ∈ 𝑉 → 𝐷 ∈ {𝐷}) | |
2 | fvres 5453 | . . . 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 4841 | . . . . 5 ⊢ ((𝐴 ∪ {〈𝐵, 𝐶〉}) ↾ {𝐷}) = ((𝐴 ↾ {𝐷}) ∪ ({〈𝐵, 𝐶〉} ↾ {𝐷})) | |
5 | elsni 3550 | . . . . . . . . 9 ⊢ (𝐵 ∈ {𝐷} → 𝐵 = 𝐷) | |
6 | 5 | necon3ai 2358 | . . . . . . . 8 ⊢ (𝐵 ≠ 𝐷 → ¬ 𝐵 ∈ {𝐷}) |
7 | ressnop0 5609 | . . . . . . . 8 ⊢ (¬ 𝐵 ∈ {𝐷} → ({〈𝐵, 𝐶〉} ↾ {𝐷}) = ∅) | |
8 | 6, 7 | syl 14 | . . . . . . 7 ⊢ (𝐵 ≠ 𝐷 → ({〈𝐵, 𝐶〉} ↾ {𝐷}) = ∅) |
9 | 8 | uneq2d 3235 | . . . . . 6 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ↾ {𝐷}) ∪ ({〈𝐵, 𝐶〉} ↾ {𝐷})) = ((𝐴 ↾ {𝐷}) ∪ ∅)) |
10 | un0 3401 | . . . . . 6 ⊢ ((𝐴 ↾ {𝐷}) ∪ ∅) = (𝐴 ↾ {𝐷}) | |
11 | 9, 10 | eqtrdi 2189 | . . . . 5 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ↾ {𝐷}) ∪ ({〈𝐵, 𝐶〉} ↾ {𝐷})) = (𝐴 ↾ {𝐷})) |
12 | 4, 11 | syl5eq 2185 | . . . 4 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ∪ {〈𝐵, 𝐶〉}) ↾ {𝐷}) = (𝐴 ↾ {𝐷})) |
13 | 12 | fveq1d 5431 | . . 3 ⊢ (𝐵 ≠ 𝐷 → (((𝐴 ∪ {〈𝐵, 𝐶〉}) ↾ {𝐷})‘𝐷) = ((𝐴 ↾ {𝐷})‘𝐷)) |
14 | 3, 13 | sylan9req 2194 | . 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → ((𝐴 ∪ {〈𝐵, 𝐶〉})‘𝐷) = ((𝐴 ↾ {𝐷})‘𝐷)) |
15 | fvres 5453 | . . . 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 2173 | 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 1332 ∈ wcel 1481 ≠ wne 2309 ∪ cun 3074 ∅c0 3368 {csn 3532 〈cop 3535 ↾ cres 4549 ‘cfv 5131 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-xp 4553 df-res 4559 df-iota 5096 df-fv 5139 |
This theorem is referenced by: fvpr1 5632 fvpr1g 5634 fvpr2g 5635 fvtp1g 5636 tfrlemisucaccv 6230 tfr1onlemsucaccv 6246 tfrcllemsucaccv 6259 ac6sfi 6800 0tonninf 10243 1tonninf 10244 hashennn 10558 zfz1isolemiso 10614 |
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