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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 6905 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴‘𝑥) = (𝐵‘𝑥)) | |
| 2 | 1 | breqd 5154 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑦(𝐴‘𝑥)𝑧 ↔ 𝑦(𝐵‘𝑥)𝑧)) |
| 3 | 2 | oprabbidv 7499 | . 2 ⊢ (𝐴 = 𝐵 → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑦(𝐴‘𝑥)𝑧} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑦(𝐵‘𝑥)𝑧}) |
| 4 | df-unc 8293 | . 2 ⊢ uncurry 𝐴 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑦(𝐴‘𝑥)𝑧} | |
| 5 | df-unc 8293 | . 2 ⊢ uncurry 𝐵 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑦(𝐵‘𝑥)𝑧} | |
| 6 | 3, 4, 5 | 3eqtr4g 2802 | 1 ⊢ (𝐴 = 𝐵 → uncurry 𝐴 = uncurry 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 class class class wbr 5143 ‘cfv 6561 {coprab 7432 uncurry cunc 8291 |
| 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-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-oprab 7435 df-unc 8293 |
| This theorem is referenced by: (None) |
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