Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mpt2difsnif | Structured version Visualization version GIF version |
Description: A mapping with two arguments with the first argument from a difference set with a singleton and a conditional as result. (Contributed by AV, 13-Feb-2019.) |
Ref | Expression |
---|---|
mpt2difsnif | ⊢ (𝑖 ∈ (𝐴 ∖ {𝑋}), 𝑗 ∈ 𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ (𝐴 ∖ {𝑋}), 𝑗 ∈ 𝐵 ↦ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifsn 4260 | . . . . 5 ⊢ (𝑖 ∈ (𝐴 ∖ {𝑋}) ↔ (𝑖 ∈ 𝐴 ∧ 𝑖 ≠ 𝑋)) | |
2 | df-ne 2782 | . . . . . . 7 ⊢ (𝑖 ≠ 𝑋 ↔ ¬ 𝑖 = 𝑋) | |
3 | 2 | biimpi 205 | . . . . . 6 ⊢ (𝑖 ≠ 𝑋 → ¬ 𝑖 = 𝑋) |
4 | 3 | adantl 481 | . . . . 5 ⊢ ((𝑖 ∈ 𝐴 ∧ 𝑖 ≠ 𝑋) → ¬ 𝑖 = 𝑋) |
5 | 1, 4 | sylbi 206 | . . . 4 ⊢ (𝑖 ∈ (𝐴 ∖ {𝑋}) → ¬ 𝑖 = 𝑋) |
6 | 5 | adantr 480 | . . 3 ⊢ ((𝑖 ∈ (𝐴 ∖ {𝑋}) ∧ 𝑗 ∈ 𝐵) → ¬ 𝑖 = 𝑋) |
7 | 6 | iffalsed 4047 | . 2 ⊢ ((𝑖 ∈ (𝐴 ∖ {𝑋}) ∧ 𝑗 ∈ 𝐵) → if(𝑖 = 𝑋, 𝐶, 𝐷) = 𝐷) |
8 | 7 | mpt2eq3ia 6618 | 1 ⊢ (𝑖 ∈ (𝐴 ∖ {𝑋}), 𝑗 ∈ 𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ (𝐴 ∖ {𝑋}), 𝑗 ∈ 𝐵 ↦ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∖ cdif 3537 ifcif 4036 {csn 4125 ↦ 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-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-v 3175 df-dif 3543 df-if 4037 df-sn 4126 df-oprab 6553 df-mpt2 6554 |
This theorem is referenced by: smadiadetglem1 20296 |
Copyright terms: Public domain | W3C validator |