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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 5917 | . . . 4 ⊢ (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵) | |
2 | 1 | dmeqd 5919 | . . 3 ⊢ (𝐴 = 𝐵 → dom dom 𝐴 = dom dom 𝐵) |
3 | breq 5150 | . . . 4 ⊢ (𝐴 = 𝐵 → (〈𝑥, 𝑦〉𝐴𝑧 ↔ 〈𝑥, 𝑦〉𝐵𝑧)) | |
4 | 3 | opabbidv 5214 | . . 3 ⊢ (𝐴 = 𝐵 → {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐴𝑧} = {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐵𝑧}) |
5 | 2, 4 | mpteq12dv 5239 | . 2 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ dom dom 𝐴 ↦ {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐴𝑧}) = (𝑥 ∈ dom dom 𝐵 ↦ {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐵𝑧})) |
6 | df-cur 8291 | . 2 ⊢ curry 𝐴 = (𝑥 ∈ dom dom 𝐴 ↦ {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐴𝑧}) | |
7 | df-cur 8291 | . 2 ⊢ curry 𝐵 = (𝑥 ∈ dom dom 𝐵 ↦ {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐵𝑧}) | |
8 | 5, 6, 7 | 3eqtr4g 2800 | 1 ⊢ (𝐴 = 𝐵 → curry 𝐴 = curry 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 〈cop 4637 class class class wbr 5148 {copab 5210 ↦ cmpt 5231 dom cdm 5689 curry ccur 8289 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-mpt 5232 df-dm 5699 df-cur 8291 |
This theorem is referenced by: curfv 37587 matunitlindf 37605 |
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