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| Mirrors > Home > MPE Home > Th. List > Mathboxes > reuabaiotaiota | Structured version Visualization version GIF version | ||
| Description: The iota and the alternate iota over a wff 𝜑 are equal iff there is a unique satisfying value of {𝑥 ∣ 𝜑} = {𝑦}. (Contributed by AV, 25-Aug-2022.) |
| Ref | Expression |
|---|---|
| reuabaiotaiota | ⊢ (∃!𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ (℩𝑥𝜑) = (℩'𝑥𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | uniintab 4953 | . 2 ⊢ (∃!𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}) | |
| 2 | df-iota 6467 | . . 3 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} | |
| 3 | df-aiota 47090 | . . 3 ⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} | |
| 4 | 2, 3 | eqeq12i 2748 | . 2 ⊢ ((℩𝑥𝜑) = (℩'𝑥𝜑) ↔ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}) |
| 5 | 1, 4 | bitr4i 278 | 1 ⊢ (∃!𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ (℩𝑥𝜑) = (℩'𝑥𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∃!weu 2562 {cab 2708 {csn 4592 ∪ cuni 4874 ∩ cint 4913 ℩cio 6465 ℩'caiota 47088 |
| 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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-sn 4593 df-pr 4595 df-uni 4875 df-int 4914 df-iota 6467 df-aiota 47090 |
| This theorem is referenced by: reuaiotaiota 47093 aiotaval 47100 |
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