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Mirrors > Home > MPE Home > Th. List > mpt2eq123i | Structured version Visualization version GIF version |
Description: An equality inference for the maps to notation. (Contributed by NM, 15-Jul-2013.) |
Ref | Expression |
---|---|
mpt2eq123i.1 | ⊢ 𝐴 = 𝐷 |
mpt2eq123i.2 | ⊢ 𝐵 = 𝐸 |
mpt2eq123i.3 | ⊢ 𝐶 = 𝐹 |
Ref | Expression |
---|---|
mpt2eq123i | ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpt2eq123i.1 | . . . 4 ⊢ 𝐴 = 𝐷 | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → 𝐴 = 𝐷) |
3 | mpt2eq123i.2 | . . . 4 ⊢ 𝐵 = 𝐸 | |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → 𝐵 = 𝐸) |
5 | mpt2eq123i.3 | . . . 4 ⊢ 𝐶 = 𝐹 | |
6 | 5 | a1i 11 | . . 3 ⊢ (⊤ → 𝐶 = 𝐹) |
7 | 2, 4, 6 | mpt2eq123dv 6615 | . 2 ⊢ (⊤ → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹)) |
8 | 7 | trud 1484 | 1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ⊤wtru 1476 ↦ cmpt2 6551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-oprab 6553 df-mpt2 6554 |
This theorem is referenced by: ofmres 7055 seqval 12674 oppgtmd 21711 mdetlap1 29220 sdc 32710 tgrpset 35051 mendvscafval 36779 fsovcnvlem 37327 hspmbl 39519 wlkson 40864 |
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