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Mirrors > Home > MPE Home > Th. List > iotaval2 | Structured version Visualization version GIF version |
Description: Version of iotaval 6534 using df-iota 6516 instead of dfiota2 6517. (Contributed by SN, 6-Nov-2024.) |
Ref | Expression |
---|---|
iotaval2 | ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iota 6516 | . 2 ⊢ (℩𝑥𝜑) = ∪ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} | |
2 | eqeq1 2739 | . . . . 5 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ({𝑥 ∣ 𝜑} = {𝑤} ↔ {𝑦} = {𝑤})) | |
3 | sneqbg 4848 | . . . . . . 7 ⊢ (𝑦 ∈ V → ({𝑦} = {𝑤} ↔ 𝑦 = 𝑤)) | |
4 | 3 | elv 3483 | . . . . . 6 ⊢ ({𝑦} = {𝑤} ↔ 𝑦 = 𝑤) |
5 | equcom 2015 | . . . . . 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 1925 | . . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∀𝑤({𝑥 ∣ 𝜑} = {𝑤} ↔ 𝑤 = 𝑦)) |
9 | uniabio 6530 | . . 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 2787 | 1 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∀wal 1535 = wceq 1537 {cab 2712 Vcvv 3478 {csn 4631 ∪ cuni 4912 ℩cio 6514 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-un 3968 df-ss 3980 df-sn 4632 df-pr 4634 df-uni 4913 df-iota 6516 |
This theorem is referenced by: iotauni2 6532 iotaval 6534 iotaex 6536 |
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