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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 5736 | . . . 4 ⊢ (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵) | |
2 | 1 | dmeqd 5738 | . . 3 ⊢ (𝐴 = 𝐵 → dom dom 𝐴 = dom dom 𝐵) |
3 | breq 5032 | . . . 4 ⊢ (𝐴 = 𝐵 → (〈𝑥, 𝑦〉𝐴𝑧 ↔ 〈𝑥, 𝑦〉𝐵𝑧)) | |
4 | 3 | opabbidv 5096 | . . 3 ⊢ (𝐴 = 𝐵 → {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐴𝑧} = {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐵𝑧}) |
5 | 2, 4 | mpteq12dv 5115 | . 2 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ dom dom 𝐴 ↦ {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐴𝑧}) = (𝑥 ∈ dom dom 𝐵 ↦ {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐵𝑧})) |
6 | df-cur 7916 | . 2 ⊢ curry 𝐴 = (𝑥 ∈ dom dom 𝐴 ↦ {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐴𝑧}) | |
7 | df-cur 7916 | . 2 ⊢ curry 𝐵 = (𝑥 ∈ dom dom 𝐵 ↦ {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐵𝑧}) | |
8 | 5, 6, 7 | 3eqtr4g 2858 | 1 ⊢ (𝐴 = 𝐵 → curry 𝐴 = curry 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 〈cop 4531 class class class wbr 5030 {copab 5092 ↦ cmpt 5110 dom cdm 5519 curry ccur 7914 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-mpt 5111 df-dm 5529 df-cur 7916 |
This theorem is referenced by: curfv 35037 matunitlindf 35055 |
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