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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 17139 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet 〈𝐴, 𝐵〉) = ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) | |
2 | resexg 6032 | . . 3 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ↾ (V ∖ {𝐴})) ∈ V) | |
3 | snex 5433 | . . . 4 ⊢ {〈𝐴, 𝐵〉} ∈ V | |
4 | 3 | a1i 11 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉} ∈ V) |
5 | unexg 7752 | . . 3 ⊢ (((𝑆 ↾ (V ∖ {𝐴})) ∈ V ∧ {〈𝐴, 𝐵〉} ∈ V) → ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉}) ∈ V) | |
6 | 2, 4, 5 | syl2an2r 683 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝑆 ↾ (V ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉}) ∈ V) |
7 | 1, 6 | eqeltrd 2825 | 1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet 〈𝐴, 𝐵〉) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 Vcvv 3461 ∖ cdif 3941 ∪ cun 3942 {csn 4630 〈cop 4636 ↾ cres 5680 (class class class)co 7419 sSet csts 17135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-res 5690 df-iota 6501 df-fun 6551 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-sets 17136 |
This theorem is referenced by: (None) |
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