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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 5545 | . 2 ⊢ (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
2 | uniexg 4228 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) | |
3 | iotass 4949 | . . . . 5 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 ⊆ ∪ 𝐴) → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) ⊆ ∪ 𝐴) | |
4 | elssuni 3655 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴) | |
5 | 4 | adantr 270 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 ⊆ ∪ 𝐴) |
6 | 3, 5 | mpg 1381 | . . . 4 ⊢ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) ⊆ ∪ 𝐴 |
7 | 6 | a1i 9 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) ⊆ ∪ 𝐴) |
8 | 2, 7 | ssexd 3944 | . 2 ⊢ (𝐴 ∈ 𝑉 → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) ∈ V) |
9 | 1, 8 | syl5eqel 2169 | 1 ⊢ (𝐴 ∈ 𝑉 → (℩𝑥 ∈ 𝐴 𝜓) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∈ wcel 1434 Vcvv 2612 ⊆ wss 2984 ∪ cuni 3627 ℩cio 4930 ℩crio 5544 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-un 4223 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-v 2614 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-uni 3628 df-iota 4932 df-riota 5545 |
This theorem is referenced by: flval 9566 sqrtrval 10258 qnumval 10941 qdenval 10942 |
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