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Mirrors > Home > MPE Home > Th. List > csbmpt2 | Structured version Visualization version GIF version |
Description: Move substitution into the second part of a maps-to notation. (Contributed by AV, 26-Sep-2019.) |
Ref | Expression |
---|---|
csbmpt2 | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑦 ∈ 𝑌 ↦ 𝑍) = (𝑦 ∈ 𝑌 ↦ ⦋𝐴 / 𝑥⦌𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbmpt12 5444 | . 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑦 ∈ 𝑌 ↦ 𝑍) = (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ↦ ⦋𝐴 / 𝑥⦌𝑍)) | |
2 | csbconstg 3902 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑌 = 𝑌) | |
3 | 2 | mpteq1d 5155 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ↦ ⦋𝐴 / 𝑥⦌𝑍) = (𝑦 ∈ 𝑌 ↦ ⦋𝐴 / 𝑥⦌𝑍)) |
4 | 1, 3 | eqtrd 2856 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑦 ∈ 𝑌 ↦ 𝑍) = (𝑦 ∈ 𝑌 ↦ ⦋𝐴 / 𝑥⦌𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ⦋csb 3883 ↦ cmpt 5146 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-opab 5129 df-mpt 5147 |
This theorem is referenced by: matgsum 21046 csbrdgg 34613 |
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