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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 6190 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴‘𝑥) = (𝐵‘𝑥)) | |
2 | 1 | breqd 4664 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑦(𝐴‘𝑥)𝑧 ↔ 𝑦(𝐵‘𝑥)𝑧)) |
3 | 2 | oprabbidv 6709 | . 2 ⊢ (𝐴 = 𝐵 → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑦(𝐴‘𝑥)𝑧} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑦(𝐵‘𝑥)𝑧}) |
4 | df-unc 7394 | . 2 ⊢ uncurry 𝐴 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑦(𝐴‘𝑥)𝑧} | |
5 | df-unc 7394 | . 2 ⊢ uncurry 𝐵 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑦(𝐵‘𝑥)𝑧} | |
6 | 3, 4, 5 | 3eqtr4g 2681 | 1 ⊢ (𝐴 = 𝐵 → uncurry 𝐴 = uncurry 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 class class class wbr 4653 ‘cfv 5888 {coprab 6651 uncurry cunc 7392 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-oprab 6654 df-unc 7394 |
This theorem is referenced by: (None) |
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