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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 2918 | . 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 7392 | 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 1540 Ⅎwnf 1784 ∈ wcel 2105 Ⅎwnfc 2882 ∃!wreu 3373 ℩crio 7367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-reu 3376 df-v 3475 df-un 3953 df-in 3955 df-ss 3965 df-sn 4629 df-pr 4631 df-uni 4909 df-iota 6495 df-riota 7368 |
This theorem is referenced by: riota2 7394 riotaprop 7396 riotass2 7399 riotass 7400 riotaxfrd 7403 ttrcltr 9717 cdlemksv2 40182 cdlemkuv2 40202 cdlemk36 40248 |
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