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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 6717 | . 2 ⊢ (⊤ → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹)) |
8 | 7 | trud 1493 | 1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ⊤wtru 1484 ↦ cmpt2 6652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-oprab 6654 df-mpt2 6655 |
This theorem is referenced by: ofmres 7164 seqval 12812 oppgtmd 21901 wlkson 26552 mdetlap1 29892 sdc 33540 tgrpset 36033 mendvscafval 37760 fsovcnvlem 38307 hspmbl 40843 |
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