Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 5468 | . 2 ⊢ (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
2 | uniexg 4175 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) | |
3 | iotass 4884 | . . . . 5 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 ⊆ ∪ 𝐴) → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) ⊆ ∪ 𝐴) | |
4 | elssuni 3608 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴) | |
5 | 4 | adantr 261 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 ⊆ ∪ 𝐴) |
6 | 3, 5 | mpg 1340 | . . . 4 ⊢ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) ⊆ ∪ 𝐴 |
7 | 6 | a1i 9 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) ⊆ ∪ 𝐴) |
8 | 2, 7 | ssexd 3897 | . 2 ⊢ (𝐴 ∈ 𝑉 → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) ∈ V) |
9 | 1, 8 | syl5eqel 2124 | 1 ⊢ (𝐴 ∈ 𝑉 → (℩𝑥 ∈ 𝐴 𝜓) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1393 Vcvv 2557 ⊆ wss 2917 ∪ cuni 3580 ℩cio 4865 ℩crio 5467 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-uni 3581 df-iota 4867 df-riota 5468 |
This theorem is referenced by: flval 9116 sqrtrval 9598 |
Copyright terms: Public domain | W3C validator |