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Mirrors > Home > MPE Home > Th. List > Mathboxes > fnxpdmdm | Structured version Visualization version GIF version |
Description: The domain of the domain of a function over a Cartesian square. (Contributed by AV, 13-Jan-2020.) |
Ref | Expression |
---|---|
fnxpdmdm | ⊢ (𝐹 Fn (𝐴 × 𝐴) → dom dom 𝐹 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fndm 6441 | . 2 ⊢ (𝐹 Fn (𝐴 × 𝐴) → dom 𝐹 = (𝐴 × 𝐴)) | |
2 | dmeq 5758 | . . 3 ⊢ (dom 𝐹 = (𝐴 × 𝐴) → dom dom 𝐹 = dom (𝐴 × 𝐴)) | |
3 | dmxpid 5786 | . . 3 ⊢ dom (𝐴 × 𝐴) = 𝐴 | |
4 | 2, 3 | syl6eq 2872 | . 2 ⊢ (dom 𝐹 = (𝐴 × 𝐴) → dom dom 𝐹 = 𝐴) |
5 | 1, 4 | syl 17 | 1 ⊢ (𝐹 Fn (𝐴 × 𝐴) → dom dom 𝐹 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 × cxp 5539 dom cdm 5541 Fn wfn 6336 |
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 2793 ax-sep 5189 ax-nul 5196 ax-pr 5316 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rab 3147 df-v 3488 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-br 5053 df-opab 5115 df-xp 5547 df-dm 5551 df-fn 6344 |
This theorem is referenced by: (None) |
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