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Mirrors > Home > MPE Home > Th. List > Mathboxes > unceq | Structured version Visualization version GIF version |
Description: Equality theorem for uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.) |
Ref | Expression |
---|---|
unceq | ⊢ (𝐴 = 𝐵 → uncurry 𝐴 = uncurry 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 6895 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴‘𝑥) = (𝐵‘𝑥)) | |
2 | 1 | breqd 5160 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑦(𝐴‘𝑥)𝑧 ↔ 𝑦(𝐵‘𝑥)𝑧)) |
3 | 2 | oprabbidv 7486 | . 2 ⊢ (𝐴 = 𝐵 → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑦(𝐴‘𝑥)𝑧} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑦(𝐵‘𝑥)𝑧}) |
4 | df-unc 8274 | . 2 ⊢ uncurry 𝐴 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑦(𝐴‘𝑥)𝑧} | |
5 | df-unc 8274 | . 2 ⊢ uncurry 𝐵 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑦(𝐵‘𝑥)𝑧} | |
6 | 3, 4, 5 | 3eqtr4g 2790 | 1 ⊢ (𝐴 = 𝐵 → uncurry 𝐴 = uncurry 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 class class class wbr 5149 ‘cfv 6549 {coprab 7420 uncurry cunc 8272 |
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 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-v 3463 df-ss 3961 df-uni 4910 df-br 5150 df-iota 6501 df-fv 6557 df-oprab 7423 df-unc 8274 |
This theorem is referenced by: (None) |
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