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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 5914 | . . . 4 ⊢ (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵) | |
| 2 | 1 | dmeqd 5916 | . . 3 ⊢ (𝐴 = 𝐵 → dom dom 𝐴 = dom dom 𝐵) |
| 3 | breq 5145 | . . . 4 ⊢ (𝐴 = 𝐵 → (〈𝑥, 𝑦〉𝐴𝑧 ↔ 〈𝑥, 𝑦〉𝐵𝑧)) | |
| 4 | 3 | opabbidv 5209 | . . 3 ⊢ (𝐴 = 𝐵 → {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐴𝑧} = {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐵𝑧}) |
| 5 | 2, 4 | mpteq12dv 5233 | . 2 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ dom dom 𝐴 ↦ {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐴𝑧}) = (𝑥 ∈ dom dom 𝐵 ↦ {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐵𝑧})) |
| 6 | df-cur 8292 | . 2 ⊢ curry 𝐴 = (𝑥 ∈ dom dom 𝐴 ↦ {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐴𝑧}) | |
| 7 | df-cur 8292 | . 2 ⊢ curry 𝐵 = (𝑥 ∈ dom dom 𝐵 ↦ {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐵𝑧}) | |
| 8 | 5, 6, 7 | 3eqtr4g 2802 | 1 ⊢ (𝐴 = 𝐵 → curry 𝐴 = curry 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 〈cop 4632 class class class wbr 5143 {copab 5205 ↦ cmpt 5225 dom cdm 5685 curry ccur 8290 |
| 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-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-mpt 5226 df-dm 5695 df-cur 8292 |
| This theorem is referenced by: curfv 37607 matunitlindf 37625 |
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