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| Mirrors > Home > MPE Home > Th. List > Mathboxes > sn-iotalemcor | Structured version Visualization version GIF version | ||
| Description: Corollary of sn-iotalem 42260. Compare sb8iota 6525. (Contributed by SN, 6-Nov-2024.) |
| Ref | Expression |
|---|---|
| sn-iotalemcor | ⊢ (℩𝑥𝜑) = (℩𝑦{𝑥 ∣ 𝜑} = {𝑦}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sn-iotalem 42260 | . . 3 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧 ∣ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧}} | |
| 2 | 1 | unieqi 4919 | . 2 ⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧}} |
| 3 | df-iota 6514 | . 2 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} | |
| 4 | df-iota 6514 | . 2 ⊢ (℩𝑦{𝑥 ∣ 𝜑} = {𝑦}) = ∪ {𝑧 ∣ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧}} | |
| 5 | 2, 3, 4 | 3eqtr4i 2775 | 1 ⊢ (℩𝑥𝜑) = (℩𝑦{𝑥 ∣ 𝜑} = {𝑦}) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 {cab 2714 {csn 4626 ∪ cuni 4907 ℩cio 6512 |
| 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 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-ss 3968 df-sn 4627 df-uni 4908 df-iota 6514 |
| This theorem is referenced by: (None) |
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