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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 2922 | . 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 7411 | 1 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 Ⅎwnfc 2890 ∃!wreu 3378 ℩crio 7387 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-reu 3381 df-v 3482 df-un 3956 df-ss 3968 df-sn 4627 df-pr 4629 df-uni 4908 df-iota 6514 df-riota 7388 |
| This theorem is referenced by: riota2 7413 riotaprop 7415 riotass2 7418 riotass 7419 riotaxfrd 7422 ttrcltr 9756 cdlemksv2 40849 cdlemkuv2 40869 cdlemk36 40915 |
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