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Mirrors > Home > ILE Home > Th. List > fnsnsplitdc | 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, 29-Jan-2023.) |
Ref | Expression |
---|---|
fnsnsplitdc | ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝐹 = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnresdm 5327 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹) | |
2 | 1 | 3ad2ant2 1019 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹 ↾ 𝐴) = 𝐹) |
3 | resundi 4922 | . . 3 ⊢ (𝐹 ↾ ((𝐴 ∖ {𝑋}) ∪ {𝑋})) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ (𝐹 ↾ {𝑋})) | |
4 | dcdifsnid 6507 | . . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑋 ∈ 𝐴) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴) | |
5 | 4 | 3adant2 1016 | . . . 4 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴) |
6 | 5 | reseq2d 4909 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹 ↾ ((𝐴 ∖ {𝑋}) ∪ {𝑋})) = (𝐹 ↾ 𝐴)) |
7 | fnressn 5704 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹 ↾ {𝑋}) = {⟨𝑋, (𝐹‘𝑋)⟩}) | |
8 | 7 | uneq2d 3291 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ (𝐹 ↾ {𝑋})) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩})) |
9 | 8 | 3adant1 1015 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ (𝐹 ↾ {𝑋})) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩})) |
10 | 3, 6, 9 | 3eqtr3a 2234 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹 ↾ 𝐴) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩})) |
11 | 2, 10 | eqtr3d 2212 | 1 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝐹 = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹‘𝑋)⟩})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 DECID wdc 834 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ∖ cdif 3128 ∪ cun 3129 {csn 3594 ⟨cop 3597 ↾ cres 4630 Fn wfn 5213 ‘cfv 5218 |
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 4123 ax-pow 4176 ax-pr 4211 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-v 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 |
This theorem is referenced by: funresdfunsndc 6509 |
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