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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 5793 | . 2 ⊢ (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
2 | uniexg 4412 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) | |
3 | iotass 5165 | . . . . 5 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 ⊆ ∪ 𝐴) → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) ⊆ ∪ 𝐴) | |
4 | elssuni 3812 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴) | |
5 | 4 | adantr 274 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 ⊆ ∪ 𝐴) |
6 | 3, 5 | mpg 1438 | . . . 4 ⊢ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) ⊆ ∪ 𝐴 |
7 | 6 | a1i 9 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) ⊆ ∪ 𝐴) |
8 | 2, 7 | ssexd 4117 | . 2 ⊢ (𝐴 ∈ 𝑉 → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) ∈ V) |
9 | 1, 8 | eqeltrid 2251 | 1 ⊢ (𝐴 ∈ 𝑉 → (℩𝑥 ∈ 𝐴 𝜓) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 2135 Vcvv 2722 ⊆ wss 3112 ∪ cuni 3784 ℩cio 5146 ℩crio 5792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-un 4406 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2724 df-un 3116 df-in 3118 df-ss 3125 df-pw 3556 df-sn 3577 df-pr 3578 df-uni 3785 df-iota 5148 df-riota 5793 |
This theorem is referenced by: flval 10198 sqrtrval 10932 qnumval 12106 qdenval 12107 |
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