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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 6664 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴‘𝑥) = (𝐵‘𝑥)) | |
2 | 1 | breqd 5070 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑦(𝐴‘𝑥)𝑧 ↔ 𝑦(𝐵‘𝑥)𝑧)) |
3 | 2 | oprabbidv 7214 | . 2 ⊢ (𝐴 = 𝐵 → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑦(𝐴‘𝑥)𝑧} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑦(𝐵‘𝑥)𝑧}) |
4 | df-unc 7928 | . 2 ⊢ uncurry 𝐴 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑦(𝐴‘𝑥)𝑧} | |
5 | df-unc 7928 | . 2 ⊢ uncurry 𝐵 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑦(𝐵‘𝑥)𝑧} | |
6 | 3, 4, 5 | 3eqtr4g 2881 | 1 ⊢ (𝐴 = 𝐵 → uncurry 𝐴 = uncurry 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 class class class wbr 5059 ‘cfv 6350 {coprab 7151 uncurry cunc 7926 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-rex 3144 df-uni 4833 df-br 5060 df-iota 6309 df-fv 6358 df-oprab 7154 df-unc 7928 |
This theorem is referenced by: (None) |
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