Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mpoeq3dv | Structured version Visualization version GIF version |
Description: An equality deduction for the maps-to notation restricted to the value of the operation. (Contributed by SO, 16-Jul-2018.) |
Ref | Expression |
---|---|
mpoeq3dv.1 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
mpoeq3dv | ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpoeq3dv.1 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
2 | 1 | 3ad2ant1 1129 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 = 𝐷) |
3 | 2 | mpoeq3dva 7231 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∈ cmpo 7158 |
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-9 2124 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-3an 1085 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-oprab 7160 df-mpo 7161 |
This theorem is referenced by: seqomeq12 8090 cantnfval 9131 seqeq2 13374 seqeq3 13375 relexpsucnnr 14384 lsmfval 18763 phssip 20802 mamuval 20997 matsc 21059 marrepval0 21170 marrepval 21171 marepvval0 21175 marepvval 21176 submaval0 21189 mdetr0 21214 mdet0 21215 mdetunilem7 21227 mdetunilem8 21228 madufval 21246 maduval 21247 maducoeval2 21249 madutpos 21251 madugsum 21252 madurid 21253 minmar1val0 21256 minmar1val 21257 pmat0opsc 21306 pmat1opsc 21307 mat2pmatval 21332 cpm2mval 21358 decpmatid 21378 pmatcollpw2lem 21385 pmatcollpw3lem 21391 mply1topmatval 21412 mp2pm2mplem1 21414 mp2pm2mplem4 21417 ttgval 26661 smatfval 31060 ofceq 31356 reprval 31881 finxpeq1 34670 matunitlindflem1 34903 idfusubc 44186 digfval 44706 |
Copyright terms: Public domain | W3C validator |