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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 6846 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴‘𝑥) = (𝐵‘𝑥)) | |
2 | 1 | breqd 5121 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑦(𝐴‘𝑥)𝑧 ↔ 𝑦(𝐵‘𝑥)𝑧)) |
3 | 2 | oprabbidv 7428 | . 2 ⊢ (𝐴 = 𝐵 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐴‘𝑥)𝑧} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐵‘𝑥)𝑧}) |
4 | df-unc 8204 | . 2 ⊢ uncurry 𝐴 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐴‘𝑥)𝑧} | |
5 | df-unc 8204 | . 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 1542 class class class wbr 5110 ‘cfv 6501 {coprab 7363 uncurry cunc 8202 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-v 3450 df-in 3922 df-ss 3932 df-uni 4871 df-br 5111 df-iota 6453 df-fv 6509 df-oprab 7366 df-unc 8204 |
This theorem is referenced by: (None) |
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