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Mirrors > Home > ILE Home > Th. List > riotaexg | GIF version |
Description: Restricted iota is a set. (Contributed by Jim Kingdon, 15-Jun-2020.) |
Ref | Expression |
---|---|
riotaexg | ⊢ (𝐴 ∈ 𝑉 → (℩𝑥 ∈ 𝐴 𝜓) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-riota 5847 | . 2 ⊢ (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
2 | uniexg 4454 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) | |
3 | iotass 5210 | . . . . 5 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 ⊆ ∪ 𝐴) → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) ⊆ ∪ 𝐴) | |
4 | elssuni 3852 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴) | |
5 | 4 | adantr 276 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 ⊆ ∪ 𝐴) |
6 | 3, 5 | mpg 1462 | . . . 4 ⊢ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) ⊆ ∪ 𝐴 |
7 | 6 | a1i 9 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) ⊆ ∪ 𝐴) |
8 | 2, 7 | ssexd 4158 | . 2 ⊢ (𝐴 ∈ 𝑉 → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) ∈ V) |
9 | 1, 8 | eqeltrid 2276 | 1 ⊢ (𝐴 ∈ 𝑉 → (℩𝑥 ∈ 𝐴 𝜓) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2160 Vcvv 2752 ⊆ wss 3144 ∪ cuni 3824 ℩cio 5191 ℩crio 5846 |
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-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-un 4448 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-uni 3825 df-iota 5193 df-riota 5847 |
This theorem is referenced by: flval 10290 sqrtrval 11027 qnumval 12203 qdenval 12204 grpidvalg 12815 fn0g 12817 grpinvval 12953 grpinvfng 12954 |
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