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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 5774 | . . . 4 ⊢ (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵) | |
2 | 1 | dmeqd 5776 | . . 3 ⊢ (𝐴 = 𝐵 → dom dom 𝐴 = dom dom 𝐵) |
3 | breq 5070 | . . . 4 ⊢ (𝐴 = 𝐵 → (〈𝑥, 𝑦〉𝐴𝑧 ↔ 〈𝑥, 𝑦〉𝐵𝑧)) | |
4 | 3 | opabbidv 5134 | . . 3 ⊢ (𝐴 = 𝐵 → {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐴𝑧} = {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐵𝑧}) |
5 | 2, 4 | mpteq12dv 5153 | . 2 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ dom dom 𝐴 ↦ {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐴𝑧}) = (𝑥 ∈ dom dom 𝐵 ↦ {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐵𝑧})) |
6 | df-cur 7935 | . 2 ⊢ curry 𝐴 = (𝑥 ∈ dom dom 𝐴 ↦ {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐴𝑧}) | |
7 | df-cur 7935 | . 2 ⊢ curry 𝐵 = (𝑥 ∈ dom dom 𝐵 ↦ {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐵𝑧}) | |
8 | 5, 6, 7 | 3eqtr4g 2883 | 1 ⊢ (𝐴 = 𝐵 → curry 𝐴 = curry 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 〈cop 4575 class class class wbr 5068 {copab 5130 ↦ cmpt 5148 dom cdm 5557 curry ccur 7933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-mpt 5149 df-dm 5567 df-cur 7935 |
This theorem is referenced by: curfv 34874 matunitlindf 34892 |
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