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| Mirrors > Home > MPE Home > Th. List > Mathboxes > xrneq1 | Structured version Visualization version GIF version | ||
| Description: Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.) |
| Ref | Expression |
|---|---|
| xrneq1 | ⊢ (𝐴 = 𝐵 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | coeq2 5869 | . . 3 ⊢ (𝐴 = 𝐵 → (◡(1st ↾ (V × V)) ∘ 𝐴) = (◡(1st ↾ (V × V)) ∘ 𝐵)) | |
| 2 | 1 | ineq1d 4219 | . 2 ⊢ (𝐴 = 𝐵 → ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐶)) = ((◡(1st ↾ (V × V)) ∘ 𝐵) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐶))) |
| 3 | df-xrn 38372 | . 2 ⊢ (𝐴 ⋉ 𝐶) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐶)) | |
| 4 | df-xrn 38372 | . 2 ⊢ (𝐵 ⋉ 𝐶) = ((◡(1st ↾ (V × V)) ∘ 𝐵) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐶)) | |
| 5 | 2, 3, 4 | 3eqtr4g 2802 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 Vcvv 3480 ∩ cin 3950 × cxp 5683 ◡ccnv 5684 ↾ cres 5687 ∘ ccom 5689 1st c1st 8012 2nd c2nd 8013 ⋉ cxrn 38181 |
| 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-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-in 3958 df-ss 3968 df-br 5144 df-opab 5206 df-co 5694 df-xrn 38372 |
| This theorem is referenced by: xrneq1i 38379 xrneq1d 38380 xrneq12 38384 |
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