![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > setsv | Structured version Visualization version GIF version |
Description: The value of the structure replacement function is a set. (Contributed by AV, 10-Nov-2021.) |
Ref | Expression |
---|---|
setsv | ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet 〈𝐴, 𝐵〉) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setsval 16210 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet 〈𝐴, 𝐵〉) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) | |
2 | resexg 5652 | . . 3 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ↾ (V ∖ {𝐴})) ∈ V) | |
3 | snex 5097 | . . . 4 ⊢ {〈𝐴, 𝐵〉} ∈ V | |
4 | 3 | a1i 11 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉} ∈ V) |
5 | unexg 7191 | . . 3 ⊢ (((𝑆 ↾ (V ∖ {𝐴})) ∈ V ∧ {〈𝐴, 𝐵〉} ∈ V) → ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉}) ∈ V) | |
6 | 2, 4, 5 | syl2an2r 676 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉}) ∈ V) |
7 | 1, 6 | eqeltrd 2876 | 1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet 〈𝐴, 𝐵〉) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∈ wcel 2157 Vcvv 3383 ∖ cdif 3764 ∪ cun 3765 {csn 4366 〈cop 4372 ↾ cres 5312 (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-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-fv 6107 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-sets 16187 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |