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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 5325 | . . . 4 ⊢ (𝐹 Fn 𝑆 → (𝐹 ↾ 𝑆) = 𝐹) | |
2 | 1 | adantr 276 | . . 3 ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → (𝐹 ↾ 𝑆) = 𝐹) |
3 | ressnop0 5697 | . . . 4 ⊢ (¬ 𝑋 ∈ 𝑆 → ({⟨𝑋, 𝑌⟩} ↾ 𝑆) = ∅) | |
4 | 3 | adantl 277 | . . 3 ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → ({⟨𝑋, 𝑌⟩} ↾ 𝑆) = ∅) |
5 | 2, 4 | uneq12d 3290 | . 2 ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → ((𝐹 ↾ 𝑆) ∪ ({⟨𝑋, 𝑌⟩} ↾ 𝑆)) = (𝐹 ∪ ∅)) |
6 | resundir 4921 | . 2 ⊢ ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ 𝑆) = ((𝐹 ↾ 𝑆) ∪ ({⟨𝑋, 𝑌⟩} ↾ 𝑆)) | |
7 | un0 3456 | . . 3 ⊢ (𝐹 ∪ ∅) = 𝐹 | |
8 | 7 | eqcomi 2181 | . 2 ⊢ 𝐹 = (𝐹 ∪ ∅) |
9 | 5, 6, 8 | 3eqtr4g 2235 | 1 ⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ 𝑆) = 𝐹) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ∪ cun 3127 ∅c0 3422 {csn 3592 ⟨cop 3595 ↾ cres 4628 Fn wfn 5211 |
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 4121 ax-pow 4174 ax-pr 4209 |
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-ral 2460 df-rex 2461 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-br 4004 df-opab 4065 df-xp 4632 df-rel 4633 df-dm 4636 df-res 4638 df-fun 5218 df-fn 5219 |
This theorem is referenced by: tfrlemisucaccv 6325 tfr1onlemsucaccv 6341 tfrcllemsucaccv 6354 |
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