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Mirrors > Home > MPE Home > Th. List > iotauni | Structured version Visualization version GIF version |
Description: Equivalence between two different forms of ℩. (Contributed by Andrew Salmon, 12-Jul-2011.) |
Ref | Expression |
---|---|
iotauni | ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-eu 2474 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) | |
2 | iotaval 5862 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧) | |
3 | uniabio 5861 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∪ {𝑥 ∣ 𝜑} = 𝑧) | |
4 | 2, 3 | eqtr4d 2659 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) |
5 | 4 | exlimiv 1858 | . 2 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) |
6 | 1, 5 | sylbi 207 | 1 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1481 = wceq 1483 ∃wex 1704 ∃!weu 2470 {cab 2608 ∪ cuni 4436 ℩cio 5849 |
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-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-rex 2918 df-v 3202 df-sbc 3436 df-un 3579 df-sn 4178 df-pr 4180 df-uni 4437 df-iota 5851 |
This theorem is referenced by: iotaint 5864 iotassuni 5867 dfiota4 5879 dfiota4OLD 5880 fveu 6183 riotauni 6617 |
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