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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 4737 | . 2 ⊢ (∃!𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}) | |
2 | df-iota 6090 | . . 3 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} | |
3 | df-aiota 41976 | . . 3 ⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} | |
4 | 2, 3 | eqeq12i 2839 | . 2 ⊢ ((℩𝑥𝜑) = (℩'𝑥𝜑) ↔ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}) |
5 | 1, 4 | bitr4i 270 | 1 ⊢ (∃!𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ (℩𝑥𝜑) = (℩'𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1656 ∃!weu 2639 {cab 2811 {csn 4399 ∪ cuni 4660 ∩ cint 4699 ℩cio 6088 ℩'caiota 41974 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-sn 4400 df-pr 4402 df-uni 4661 df-int 4700 df-iota 6090 df-aiota 41976 |
This theorem is referenced by: reuaiotaiota 41979 aiotaval 41984 |
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