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Mirrors > Home > MPE Home > Th. List > riota2f | Structured version Visualization version GIF version |
Description: This theorem shows a condition that allows to represent a descriptor with a class expression 𝐵. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Ref | Expression |
---|---|
riota2f.1 | ⊢ Ⅎ𝑥𝐵 |
riota2f.2 | ⊢ Ⅎ𝑥𝜓 |
riota2f.3 | ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
riota2f | ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | riota2f.1 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
2 | 1 | nfel1 2913 | . 2 ⊢ Ⅎ𝑥 𝐵 ∈ 𝐴 |
3 | 1 | a1i 11 | . 2 ⊢ (𝐵 ∈ 𝐴 → Ⅎ𝑥𝐵) |
4 | riota2f.2 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
5 | 4 | a1i 11 | . 2 ⊢ (𝐵 ∈ 𝐴 → Ⅎ𝑥𝜓) |
6 | id 22 | . 2 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ 𝐴) | |
7 | riota2f.3 | . . 3 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) | |
8 | 7 | adantl 481 | . 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝑥 = 𝐵) → (𝜑 ↔ 𝜓)) |
9 | 2, 3, 5, 6, 8 | riota2df 7384 | 1 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 Ⅎwnfc 2877 ∃!wreu 3368 ℩crio 7359 |
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 2163 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-reu 3371 df-v 3470 df-un 3948 df-in 3950 df-ss 3960 df-sn 4624 df-pr 4626 df-uni 4903 df-iota 6488 df-riota 7360 |
This theorem is referenced by: riota2 7386 riotaprop 7388 riotass2 7391 riotass 7392 riotaxfrd 7395 ttrcltr 9710 cdlemksv2 40230 cdlemkuv2 40250 cdlemk36 40296 |
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