Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mpoeq123i | Structured version Visualization version GIF version |
Description: An equality inference for the maps-to notation. (Contributed by NM, 15-Jul-2013.) |
Ref | Expression |
---|---|
mpoeq123i.1 | ⊢ 𝐴 = 𝐷 |
mpoeq123i.2 | ⊢ 𝐵 = 𝐸 |
mpoeq123i.3 | ⊢ 𝐶 = 𝐹 |
Ref | Expression |
---|---|
mpoeq123i | ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpoeq123i.1 | . . . 4 ⊢ 𝐴 = 𝐷 | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → 𝐴 = 𝐷) |
3 | mpoeq123i.2 | . . . 4 ⊢ 𝐵 = 𝐸 | |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → 𝐵 = 𝐸) |
5 | mpoeq123i.3 | . . . 4 ⊢ 𝐶 = 𝐹 | |
6 | 5 | a1i 11 | . . 3 ⊢ (⊤ → 𝐶 = 𝐹) |
7 | 2, 4, 6 | mpoeq123dv 7412 | . 2 ⊢ (⊤ → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹)) |
8 | 7 | mptru 1547 | 1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ⊤wtru 1541 ∈ cmpo 7339 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-oprab 7341 df-mpo 7342 |
This theorem is referenced by: ofmres 7895 seqval 13833 oppgtmd 23354 wlkson 28312 mdetlap1 32074 sdc 36015 tgrpset 39021 mendvscafval 41286 fsovcnvlem 41950 hspmbl 44512 |
Copyright terms: Public domain | W3C validator |