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Mirrors > Home > ILE Home > Th. List > funresdfunsnss | GIF version |
Description: Restricting a function to a domain without one element of the domain of the function, and adding a pair of this element and the function value of the element results in a subset of the function itself. (Contributed by AV, 2-Dec-2018.) (Revised by Jim Kingdon, 21-Jan-2023.) |
Ref | Expression |
---|---|
funresdfunsnss | ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ((𝐹 ↾ (V ∖ {𝑋})) ∪ {〈𝑋, (𝐹‘𝑋)〉}) ⊆ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funrel 5199 | . . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹) | |
2 | resdmdfsn 4921 | . . . . 5 ⊢ (Rel 𝐹 → (𝐹 ↾ (V ∖ {𝑋})) = (𝐹 ↾ (dom 𝐹 ∖ {𝑋}))) | |
3 | 1, 2 | syl 14 | . . . 4 ⊢ (Fun 𝐹 → (𝐹 ↾ (V ∖ {𝑋})) = (𝐹 ↾ (dom 𝐹 ∖ {𝑋}))) |
4 | 3 | adantr 274 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → (𝐹 ↾ (V ∖ {𝑋})) = (𝐹 ↾ (dom 𝐹 ∖ {𝑋}))) |
5 | 4 | uneq1d 3270 | . 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ((𝐹 ↾ (V ∖ {𝑋})) ∪ {〈𝑋, (𝐹‘𝑋)〉}) = ((𝐹 ↾ (dom 𝐹 ∖ {𝑋})) ∪ {〈𝑋, (𝐹‘𝑋)〉})) |
6 | funfn 5212 | . . 3 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹) | |
7 | fnsnsplitss 5678 | . . 3 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ((𝐹 ↾ (dom 𝐹 ∖ {𝑋})) ∪ {〈𝑋, (𝐹‘𝑋)〉}) ⊆ 𝐹) | |
8 | 6, 7 | sylanb 282 | . 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ((𝐹 ↾ (dom 𝐹 ∖ {𝑋})) ∪ {〈𝑋, (𝐹‘𝑋)〉}) ⊆ 𝐹) |
9 | 5, 8 | eqsstrd 3173 | 1 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ((𝐹 ↾ (V ∖ {𝑋})) ∪ {〈𝑋, (𝐹‘𝑋)〉}) ⊆ 𝐹) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 Vcvv 2721 ∖ cdif 3108 ∪ cun 3109 ⊆ wss 3111 {csn 3570 〈cop 3573 dom cdm 4598 ↾ cres 4600 Rel wrel 4603 Fun wfun 5176 Fn wfn 5177 ‘cfv 5182 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 |
This theorem is referenced by: (None) |
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