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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 3447 | . . . 4 ⊢ (𝐷 ∈ 𝑉 → 𝐷 ∈ {𝐷}) | |
2 | fvres 5274 | . . . 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 4685 | . . . . 5 ⊢ ((𝐴 ∪ {〈𝐵, 𝐶〉}) ↾ {𝐷}) = ((𝐴 ↾ {𝐷}) ∪ ({〈𝐵, 𝐶〉} ↾ {𝐷})) | |
5 | elsni 3440 | . . . . . . . . 9 ⊢ (𝐵 ∈ {𝐷} → 𝐵 = 𝐷) | |
6 | 5 | necon3ai 2298 | . . . . . . . 8 ⊢ (𝐵 ≠ 𝐷 → ¬ 𝐵 ∈ {𝐷}) |
7 | ressnop0 5420 | . . . . . . . 8 ⊢ (¬ 𝐵 ∈ {𝐷} → ({〈𝐵, 𝐶〉} ↾ {𝐷}) = ∅) | |
8 | 6, 7 | syl 14 | . . . . . . 7 ⊢ (𝐵 ≠ 𝐷 → ({〈𝐵, 𝐶〉} ↾ {𝐷}) = ∅) |
9 | 8 | uneq2d 3138 | . . . . . 6 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ↾ {𝐷}) ∪ ({〈𝐵, 𝐶〉} ↾ {𝐷})) = ((𝐴 ↾ {𝐷}) ∪ ∅)) |
10 | un0 3299 | . . . . . 6 ⊢ ((𝐴 ↾ {𝐷}) ∪ ∅) = (𝐴 ↾ {𝐷}) | |
11 | 9, 10 | syl6eq 2131 | . . . . 5 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ↾ {𝐷}) ∪ ({〈𝐵, 𝐶〉} ↾ {𝐷})) = (𝐴 ↾ {𝐷})) |
12 | 4, 11 | syl5eq 2127 | . . . 4 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ∪ {〈𝐵, 𝐶〉}) ↾ {𝐷}) = (𝐴 ↾ {𝐷})) |
13 | 12 | fveq1d 5255 | . . 3 ⊢ (𝐵 ≠ 𝐷 → (((𝐴 ∪ {〈𝐵, 𝐶〉}) ↾ {𝐷})‘𝐷) = ((𝐴 ↾ {𝐷})‘𝐷)) |
14 | 3, 13 | sylan9req 2136 | . 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → ((𝐴 ∪ {〈𝐵, 𝐶〉})‘𝐷) = ((𝐴 ↾ {𝐷})‘𝐷)) |
15 | fvres 5274 | . . . 4 ⊢ (𝐷 ∈ {𝐷} → ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴‘𝐷)) | |
16 | 1, 15 | syl 14 | . . 3 ⊢ (𝐷 ∈ 𝑉 → ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴‘𝐷)) |
17 | 16 | adantr 270 | . 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴‘𝐷)) |
18 | 14, 17 | eqtrd 2115 | 1 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → ((𝐴 ∪ {〈𝐵, 𝐶〉})‘𝐷) = (𝐴‘𝐷)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 = wceq 1285 ∈ wcel 1434 ≠ wne 2249 ∪ cun 2982 ∅c0 3269 {csn 3422 〈cop 3425 ↾ cres 4403 ‘cfv 4969 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 4000 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-v 2614 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-br 3812 df-opab 3866 df-xp 4407 df-res 4413 df-iota 4934 df-fv 4977 |
This theorem is referenced by: fvpr1 5441 fvpr1g 5443 fvpr2g 5444 fvtp1g 5445 tfrlemisucaccv 6022 tfr1onlemsucaccv 6038 tfrcllemsucaccv 6051 ac6sfi 6544 0tonninf 9734 1tonninf 9735 hashennn 10023 |
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