Proof of Theorem xpider
| Step | Hyp | Ref
| Expression |
| 1 | | relxp 5703 |
. 2
⊢ Rel
(𝐴 × 𝐴) |
| 2 | | dmxpid 5941 |
. 2
⊢ dom
(𝐴 × 𝐴) = 𝐴 |
| 3 | | cnvxp 6177 |
. . 3
⊢ ◡(𝐴 × 𝐴) = (𝐴 × 𝐴) |
| 4 | | xpidtr 6142 |
. . 3
⊢ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) |
| 5 | | uneq1 4161 |
. . . 4
⊢ (◡(𝐴 × 𝐴) = (𝐴 × 𝐴) → (◡(𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴))) |
| 6 | | unss2 4187 |
. . . 4
⊢ (((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) → (◡(𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (◡(𝐴 × 𝐴) ∪ (𝐴 × 𝐴))) |
| 7 | | unidm 4157 |
. . . . 5
⊢ ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = (𝐴 × 𝐴) |
| 8 | | eqtr 2760 |
. . . . . 6
⊢ (((◡(𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) ∧ ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = (𝐴 × 𝐴)) → (◡(𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = (𝐴 × 𝐴)) |
| 9 | | sseq2 4010 |
. . . . . . 7
⊢ ((◡(𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = (𝐴 × 𝐴) → ((◡(𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (◡(𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) ↔ (◡(𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (𝐴 × 𝐴))) |
| 10 | 9 | biimpd 229 |
. . . . . 6
⊢ ((◡(𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = (𝐴 × 𝐴) → ((◡(𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (◡(𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) → (◡(𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (𝐴 × 𝐴))) |
| 11 | 8, 10 | syl 17 |
. . . . 5
⊢ (((◡(𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) ∧ ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = (𝐴 × 𝐴)) → ((◡(𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (◡(𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) → (◡(𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (𝐴 × 𝐴))) |
| 12 | 7, 11 | mpan2 691 |
. . . 4
⊢ ((◡(𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) → ((◡(𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (◡(𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) → (◡(𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (𝐴 × 𝐴))) |
| 13 | 5, 6, 12 | syl2im 40 |
. . 3
⊢ (◡(𝐴 × 𝐴) = (𝐴 × 𝐴) → (((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) → (◡(𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (𝐴 × 𝐴))) |
| 14 | 3, 4, 13 | mp2 9 |
. 2
⊢ (◡(𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (𝐴 × 𝐴) |
| 15 | | df-er 8745 |
. 2
⊢ ((𝐴 × 𝐴) Er 𝐴 ↔ (Rel (𝐴 × 𝐴) ∧ dom (𝐴 × 𝐴) = 𝐴 ∧ (◡(𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (𝐴 × 𝐴))) |
| 16 | 1, 2, 14, 15 | mpbir3an 1342 |
1
⊢ (𝐴 × 𝐴) Er 𝐴 |