![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > iotauni2 | Structured version Visualization version GIF version |
Description: Version of iotauni 6528 using df-iota 6505 instead of dfiota2 6506. (Contributed by SN, 6-Nov-2024.) |
Ref | Expression |
---|---|
iotauni2 | ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iotaval2 6521 | . . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦) | |
2 | unieq 4923 | . . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∪ {𝑥 ∣ 𝜑} = ∪ {𝑦}) | |
3 | unisnv 4934 | . . . 4 ⊢ ∪ {𝑦} = 𝑦 | |
4 | 2, 3 | eqtr2di 2785 | . . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → 𝑦 = ∪ {𝑥 ∣ 𝜑}) |
5 | 1, 4 | eqtrd 2768 | . 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) |
6 | 5 | exlimiv 1925 | 1 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∃wex 1773 {cab 2705 {csn 4632 ∪ cuni 4912 ℩cio 6503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3475 df-un 3954 df-in 3956 df-ss 3966 df-sn 4633 df-pr 4635 df-uni 4913 df-iota 6505 |
This theorem is referenced by: iotassuni 6525 |
Copyright terms: Public domain | W3C validator |