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Mirrors > Home > MPE Home > Th. List > Mathboxes > sn-iotalemcor | Structured version Visualization version GIF version |
Description: Corollary of sn-iotalem 40176. Compare sb8iota 6398. (Contributed by SN, 6-Nov-2024.) |
Ref | Expression |
---|---|
sn-iotalemcor | ⊢ (℩𝑥𝜑) = (℩𝑦{𝑥 ∣ 𝜑} = {𝑦}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sn-iotalem 40176 | . . 3 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧 ∣ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧}} | |
2 | 1 | unieqi 4854 | . 2 ⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧}} |
3 | df-iota 6386 | . 2 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} | |
4 | df-iota 6386 | . 2 ⊢ (℩𝑦{𝑥 ∣ 𝜑} = {𝑦}) = ∪ {𝑧 ∣ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧}} | |
5 | 2, 3, 4 | 3eqtr4i 2776 | 1 ⊢ (℩𝑥𝜑) = (℩𝑦{𝑥 ∣ 𝜑} = {𝑦}) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 {cab 2715 {csn 4563 ∪ cuni 4841 ℩cio 6384 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-v 3433 df-in 3895 df-ss 3905 df-sn 4564 df-uni 4842 df-iota 6386 |
This theorem is referenced by: (None) |
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