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Mirrors > Home > MPE Home > Th. List > fvsetsid | Structured version Visualization version GIF version |
Description: The value of the structure replacement function for its first argument is its second argument. (Contributed by SO, 12-Jul-2018.) |
Ref | Expression |
---|---|
fvsetsid | ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → ((𝐹 sSet 〈𝑋, 𝑌〉)‘𝑋) = 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setsval 16210 | . . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑌 ∈ 𝑈) → (𝐹 sSet 〈𝑋, 𝑌〉) = ((𝐹 ↾ (V ∖ {𝑋})) ∪ {〈𝑋, 𝑌〉})) | |
2 | 1 | 3adant2 1162 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → (𝐹 sSet 〈𝑋, 𝑌〉) = ((𝐹 ↾ (V ∖ {𝑋})) ∪ {〈𝑋, 𝑌〉})) |
3 | 2 | fveq1d 6411 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → ((𝐹 sSet 〈𝑋, 𝑌〉)‘𝑋) = (((𝐹 ↾ (V ∖ {𝑋})) ∪ {〈𝑋, 𝑌〉})‘𝑋)) |
4 | simp2 1168 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → 𝑋 ∈ 𝑊) | |
5 | simp3 1169 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → 𝑌 ∈ 𝑈) | |
6 | neldifsn 4509 | . . . . 5 ⊢ ¬ 𝑋 ∈ (V ∖ {𝑋}) | |
7 | dmres 5627 | . . . . . . 7 ⊢ dom (𝐹 ↾ (V ∖ {𝑋})) = ((V ∖ {𝑋}) ∩ dom 𝐹) | |
8 | inss1 4026 | . . . . . . 7 ⊢ ((V ∖ {𝑋}) ∩ dom 𝐹) ⊆ (V ∖ {𝑋}) | |
9 | 7, 8 | eqsstri 3829 | . . . . . 6 ⊢ dom (𝐹 ↾ (V ∖ {𝑋})) ⊆ (V ∖ {𝑋}) |
10 | 9 | sseli 3792 | . . . . 5 ⊢ (𝑋 ∈ dom (𝐹 ↾ (V ∖ {𝑋})) → 𝑋 ∈ (V ∖ {𝑋})) |
11 | 6, 10 | mto 189 | . . . 4 ⊢ ¬ 𝑋 ∈ dom (𝐹 ↾ (V ∖ {𝑋})) |
12 | 11 | a1i 11 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → ¬ 𝑋 ∈ dom (𝐹 ↾ (V ∖ {𝑋}))) |
13 | fsnunfv 6680 | . . 3 ⊢ ((𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈 ∧ ¬ 𝑋 ∈ dom (𝐹 ↾ (V ∖ {𝑋}))) → (((𝐹 ↾ (V ∖ {𝑋})) ∪ {〈𝑋, 𝑌〉})‘𝑋) = 𝑌) | |
14 | 4, 5, 12, 13 | syl3anc 1491 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → (((𝐹 ↾ (V ∖ {𝑋})) ∪ {〈𝑋, 𝑌〉})‘𝑋) = 𝑌) |
15 | 3, 14 | eqtrd 2831 | 1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → ((𝐹 sSet 〈𝑋, 𝑌〉)‘𝑋) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 Vcvv 3383 ∖ cdif 3764 ∪ cun 3765 ∩ cin 3766 {csn 4366 〈cop 4372 dom cdm 5310 ↾ cres 5312 ‘cfv 6099 (class class class)co 6876 sSet csts 16178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2375 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pr 5095 ax-un 7181 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-ral 3092 df-rex 3093 df-rab 3096 df-v 3385 df-sbc 3632 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-br 4842 df-opab 4904 df-id 5218 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-res 5322 df-iota 6062 df-fun 6101 df-fn 6102 df-fv 6107 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-sets 16187 |
This theorem is referenced by: mdetunilem9 20748 |
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