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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 5731 | . . 3 ⊢ (𝐴 = 𝐵 → (◡(1st ↾ (V × V)) ∘ 𝐴) = (◡(1st ↾ (V × V)) ∘ 𝐵)) | |
2 | 1 | ineq1d 4190 | . 2 ⊢ (𝐴 = 𝐵 → ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐶)) = ((◡(1st ↾ (V × V)) ∘ 𝐵) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐶))) |
3 | df-xrn 35625 | . 2 ⊢ (𝐴 ⋉ 𝐶) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐶)) | |
4 | df-xrn 35625 | . 2 ⊢ (𝐵 ⋉ 𝐶) = ((◡(1st ↾ (V × V)) ∘ 𝐵) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐶)) | |
5 | 2, 3, 4 | 3eqtr4g 2883 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 Vcvv 3496 ∩ cin 3937 × cxp 5555 ◡ccnv 5556 ↾ cres 5559 ∘ ccom 5561 1st c1st 7689 2nd c2nd 7690 ⋉ cxrn 35454 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-in 3945 df-ss 3954 df-br 5069 df-opab 5131 df-co 5566 df-xrn 35625 |
This theorem is referenced by: xrneq1i 35632 xrneq1d 35633 xrneq12 35637 |
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