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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 5569 | . . . 4 ⊢ (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵) | |
2 | 1 | dmeqd 5571 | . . 3 ⊢ (𝐴 = 𝐵 → dom dom 𝐴 = dom dom 𝐵) |
3 | breq 4888 | . . . 4 ⊢ (𝐴 = 𝐵 → (〈𝑥, 𝑦〉𝐴𝑧 ↔ 〈𝑥, 𝑦〉𝐵𝑧)) | |
4 | 3 | opabbidv 4952 | . . 3 ⊢ (𝐴 = 𝐵 → {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐴𝑧} = {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐵𝑧}) |
5 | 2, 4 | mpteq12dv 4969 | . 2 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ dom dom 𝐴 ↦ {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐴𝑧}) = (𝑥 ∈ dom dom 𝐵 ↦ {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐵𝑧})) |
6 | df-cur 7675 | . 2 ⊢ curry 𝐴 = (𝑥 ∈ dom dom 𝐴 ↦ {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐴𝑧}) | |
7 | df-cur 7675 | . 2 ⊢ curry 𝐵 = (𝑥 ∈ dom dom 𝐵 ↦ {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐵𝑧}) | |
8 | 5, 6, 7 | 3eqtr4g 2838 | 1 ⊢ (𝐴 = 𝐵 → curry 𝐴 = curry 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 〈cop 4403 class class class wbr 4886 {copab 4948 ↦ cmpt 4965 dom cdm 5355 curry ccur 7673 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ral 3094 df-rab 3098 df-v 3399 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-br 4887 df-opab 4949 df-mpt 4966 df-dm 5365 df-cur 7675 |
This theorem is referenced by: curfv 34008 matunitlindf 34027 |
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