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Mirrors > Home > ILE Home > Th. List > fnsnsplitss | GIF version |
Description: Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by Jim Kingdon, 20-Jan-2023.) |
Ref | Expression |
---|---|
fnsnsplitss | ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {〈𝑋, (𝐹‘𝑋)〉}) ⊆ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difsnss 3605 | . . . 4 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ⊆ 𝐴) | |
2 | 1 | adantl 272 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ⊆ 𝐴) |
3 | ssres2 4772 | . . 3 ⊢ (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ⊆ 𝐴 → (𝐹 ↾ ((𝐴 ∖ {𝑋}) ∪ {𝑋})) ⊆ (𝐹 ↾ 𝐴)) | |
4 | 2, 3 | syl 14 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹 ↾ ((𝐴 ∖ {𝑋}) ∪ {𝑋})) ⊆ (𝐹 ↾ 𝐴)) |
5 | resundi 4758 | . . 3 ⊢ (𝐹 ↾ ((𝐴 ∖ {𝑋}) ∪ {𝑋})) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ (𝐹 ↾ {𝑋})) | |
6 | fnressn 5522 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹 ↾ {𝑋}) = {〈𝑋, (𝐹‘𝑋)〉}) | |
7 | 6 | uneq2d 3169 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ (𝐹 ↾ {𝑋})) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {〈𝑋, (𝐹‘𝑋)〉})) |
8 | 5, 7 | syl5eq 2139 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹 ↾ ((𝐴 ∖ {𝑋}) ∪ {𝑋})) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {〈𝑋, (𝐹‘𝑋)〉})) |
9 | fnresdm 5157 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹) | |
10 | 9 | adantr 271 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹 ↾ 𝐴) = 𝐹) |
11 | 4, 8, 10 | 3sstr3d 3083 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {〈𝑋, (𝐹‘𝑋)〉}) ⊆ 𝐹) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1296 ∈ wcel 1445 ∖ cdif 3010 ∪ cun 3011 ⊆ wss 3013 {csn 3466 〈cop 3469 ↾ cres 4469 Fn wfn 5044 ‘cfv 5049 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-reu 2377 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 |
This theorem is referenced by: funresdfunsnss 5539 |
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