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Mirrors > Home > MPE Home > Th. List > Mathboxes > cureq | Structured version Visualization version GIF version |
Description: Equality theorem for currying. (Contributed by Brendan Leahy, 2-Jun-2021.) |
Ref | Expression |
---|---|
cureq | ⊢ (𝐴 = 𝐵 → curry 𝐴 = curry 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmeq 5857 | . . . 4 ⊢ (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵) | |
2 | 1 | dmeqd 5859 | . . 3 ⊢ (𝐴 = 𝐵 → dom dom 𝐴 = dom dom 𝐵) |
3 | breq 5105 | . . . 4 ⊢ (𝐴 = 𝐵 → (⟨𝑥, 𝑦⟩𝐴𝑧 ↔ ⟨𝑥, 𝑦⟩𝐵𝑧)) | |
4 | 3 | opabbidv 5169 | . . 3 ⊢ (𝐴 = 𝐵 → {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐴𝑧} = {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐵𝑧}) |
5 | 2, 4 | mpteq12dv 5194 | . 2 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ dom dom 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐴𝑧}) = (𝑥 ∈ dom dom 𝐵 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐵𝑧})) |
6 | df-cur 8194 | . 2 ⊢ curry 𝐴 = (𝑥 ∈ dom dom 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐴𝑧}) | |
7 | df-cur 8194 | . 2 ⊢ curry 𝐵 = (𝑥 ∈ dom dom 𝐵 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐵𝑧}) | |
8 | 5, 6, 7 | 3eqtr4g 2801 | 1 ⊢ (𝐴 = 𝐵 → curry 𝐴 = curry 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ⟨cop 4590 class class class wbr 5103 {copab 5165 ↦ cmpt 5186 dom cdm 5631 curry ccur 8192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-br 5104 df-opab 5166 df-mpt 5187 df-dm 5641 df-cur 8194 |
This theorem is referenced by: curfv 36025 matunitlindf 36043 |
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