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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 3622 | . . . 4 ⊢ (𝐷 ∈ 𝑉 → 𝐷 ∈ {𝐷}) | |
2 | fvres 5540 | . . . 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 4922 | . . . . 5 ⊢ ((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷}) = ((𝐴 ↾ {𝐷}) ∪ ({⟨𝐵, 𝐶⟩} ↾ {𝐷})) | |
5 | elsni 3611 | . . . . . . . . 9 ⊢ (𝐵 ∈ {𝐷} → 𝐵 = 𝐷) | |
6 | 5 | necon3ai 2396 | . . . . . . . 8 ⊢ (𝐵 ≠ 𝐷 → ¬ 𝐵 ∈ {𝐷}) |
7 | ressnop0 5698 | . . . . . . . 8 ⊢ (¬ 𝐵 ∈ {𝐷} → ({⟨𝐵, 𝐶⟩} ↾ {𝐷}) = ∅) | |
8 | 6, 7 | syl 14 | . . . . . . 7 ⊢ (𝐵 ≠ 𝐷 → ({⟨𝐵, 𝐶⟩} ↾ {𝐷}) = ∅) |
9 | 8 | uneq2d 3290 | . . . . . 6 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ↾ {𝐷}) ∪ ({⟨𝐵, 𝐶⟩} ↾ {𝐷})) = ((𝐴 ↾ {𝐷}) ∪ ∅)) |
10 | un0 3457 | . . . . . 6 ⊢ ((𝐴 ↾ {𝐷}) ∪ ∅) = (𝐴 ↾ {𝐷}) | |
11 | 9, 10 | eqtrdi 2226 | . . . . 5 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ↾ {𝐷}) ∪ ({⟨𝐵, 𝐶⟩} ↾ {𝐷})) = (𝐴 ↾ {𝐷})) |
12 | 4, 11 | eqtrid 2222 | . . . 4 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷}) = (𝐴 ↾ {𝐷})) |
13 | 12 | fveq1d 5518 | . . 3 ⊢ (𝐵 ≠ 𝐷 → (((𝐴 ∪ {⟨𝐵, 𝐶⟩}) ↾ {𝐷})‘𝐷) = ((𝐴 ↾ {𝐷})‘𝐷)) |
14 | 3, 13 | sylan9req 2231 | . 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷) = ((𝐴 ↾ {𝐷})‘𝐷)) |
15 | fvres 5540 | . . . 4 ⊢ (𝐷 ∈ {𝐷} → ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴‘𝐷)) | |
16 | 1, 15 | syl 14 | . . 3 ⊢ (𝐷 ∈ 𝑉 → ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴‘𝐷)) |
17 | 16 | adantr 276 | . 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴‘𝐷)) |
18 | 14, 17 | eqtrd 2210 | 1 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷) = (𝐴‘𝐷)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 ∪ cun 3128 ∅c0 3423 {csn 3593 ⟨cop 3596 ↾ cres 4629 ‘cfv 5217 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-pow 4175 ax-pr 4210 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-v 2740 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-br 4005 df-opab 4066 df-xp 4633 df-res 4639 df-iota 5179 df-fv 5225 |
This theorem is referenced by: fvpr1 5721 fvpr1g 5723 fvpr2g 5724 fvtp1g 5725 tfrlemisucaccv 6326 tfr1onlemsucaccv 6342 tfrcllemsucaccv 6355 ac6sfi 6898 0tonninf 10439 1tonninf 10440 hashennn 10760 zfz1isolemiso 10819 |
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