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| Mirrors > Home > MPE Home > Th. List > iotaval2 | Structured version Visualization version GIF version | ||
| Description: Version of iotaval 6532 using df-iota 6514 instead of dfiota2 6515. (Contributed by SN, 6-Nov-2024.) |
| Ref | Expression |
|---|---|
| iotaval2 | ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-iota 6514 | . 2 ⊢ (℩𝑥𝜑) = ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} | |
| 2 | eqeq1 2741 | . . . . 5 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ({𝑥 ∣ 𝜑} = {𝑤} ↔ {𝑦} = {𝑤})) | |
| 3 | sneqbg 4843 | . . . . . . 7 ⊢ (𝑦 ∈ V → ({𝑦} = {𝑤} ↔ 𝑦 = 𝑤)) | |
| 4 | 3 | elv 3485 | . . . . . 6 ⊢ ({𝑦} = {𝑤} ↔ 𝑦 = 𝑤) |
| 5 | equcom 2017 | . . . . . 6 ⊢ (𝑦 = 𝑤 ↔ 𝑤 = 𝑦) | |
| 6 | 4, 5 | bitri 275 | . . . . 5 ⊢ ({𝑦} = {𝑤} ↔ 𝑤 = 𝑦) |
| 7 | 2, 6 | bitrdi 287 | . . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ({𝑥 ∣ 𝜑} = {𝑤} ↔ 𝑤 = 𝑦)) |
| 8 | 7 | alrimiv 1927 | . . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∀𝑤({𝑥 ∣ 𝜑} = {𝑤} ↔ 𝑤 = 𝑦)) |
| 9 | uniabio 6528 | . . 3 ⊢ (∀𝑤({𝑥 ∣ 𝜑} = {𝑤} ↔ 𝑤 = 𝑦) → ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} = 𝑦) | |
| 10 | 8, 9 | syl 17 | . 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} = 𝑦) |
| 11 | 1, 10 | eqtrid 2789 | 1 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 = wceq 1540 {cab 2714 Vcvv 3480 {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-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-ss 3968 df-sn 4627 df-pr 4629 df-uni 4908 df-iota 6514 |
| This theorem is referenced by: iotauni2 6530 iotaval 6532 iotaex 6534 |
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