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Mirrors > Home > ILE Home > Th. List > fsnunres | GIF version |
Description: Recover the original function from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
Ref | Expression |
---|---|
fsnunres | ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → ((𝐹 ∪ {〈𝑋, 𝑌〉}) ↾ 𝑆) = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnresdm 5227 | . . . 4 ⊢ (𝐹 Fn 𝑆 → (𝐹 ↾ 𝑆) = 𝐹) | |
2 | 1 | adantr 274 | . . 3 ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → (𝐹 ↾ 𝑆) = 𝐹) |
3 | ressnop0 5594 | . . . 4 ⊢ (¬ 𝑋 ∈ 𝑆 → ({〈𝑋, 𝑌〉} ↾ 𝑆) = ∅) | |
4 | 3 | adantl 275 | . . 3 ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → ({〈𝑋, 𝑌〉} ↾ 𝑆) = ∅) |
5 | 2, 4 | uneq12d 3226 | . 2 ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → ((𝐹 ↾ 𝑆) ∪ ({〈𝑋, 𝑌〉} ↾ 𝑆)) = (𝐹 ∪ ∅)) |
6 | resundir 4828 | . 2 ⊢ ((𝐹 ∪ {〈𝑋, 𝑌〉}) ↾ 𝑆) = ((𝐹 ↾ 𝑆) ∪ ({〈𝑋, 𝑌〉} ↾ 𝑆)) | |
7 | un0 3391 | . . 3 ⊢ (𝐹 ∪ ∅) = 𝐹 | |
8 | 7 | eqcomi 2141 | . 2 ⊢ 𝐹 = (𝐹 ∪ ∅) |
9 | 5, 6, 8 | 3eqtr4g 2195 | 1 ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → ((𝐹 ∪ {〈𝑋, 𝑌〉}) ↾ 𝑆) = 𝐹) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ∪ cun 3064 ∅c0 3358 {csn 3522 〈cop 3525 ↾ cres 4536 Fn wfn 5113 |
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-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-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-dm 4544 df-res 4546 df-fun 5120 df-fn 5121 |
This theorem is referenced by: tfrlemisucaccv 6215 tfr1onlemsucaccv 6231 tfrcllemsucaccv 6244 |
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