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| 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 7508 | . 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 7433 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-oprab 7435 df-mpo 7436 | 
| This theorem is referenced by: ofmres 8009 seqval 14053 oppgtmd 24105 seqsval 28294 wlkson 29674 mdetlap1 33825 sdc 37751 tgrpset 40747 mendvscafval 43198 fsovcnvlem 44026 hspmbl 46644 | 
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