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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 us 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 2779 | . 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 482 | . 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝑥 = 𝐵) → (𝜑 ↔ 𝜓)) |
9 | 2, 3, 5, 6, 8 | riota2df 6631 | 1 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 Ⅎwnfc 2751 ∃!wreu 2914 ℩crio 6610 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-reu 2919 df-v 3202 df-sbc 3436 df-un 3579 df-sn 4178 df-pr 4180 df-uni 4437 df-iota 5851 df-riota 6611 |
This theorem is referenced by: riota2 6633 riotaprop 6635 riotass2 6638 riotass 6639 riotaxfrd 6642 cdlemksv2 36135 cdlemkuv2 36155 cdlemk36 36201 |
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