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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 6481 | . 2 ⊢ (𝐹 Fn (𝐴 × 𝐴) → dom 𝐹 = (𝐴 × 𝐴)) | |
2 | dmeq 5772 | . . 3 ⊢ (dom 𝐹 = (𝐴 × 𝐴) → dom dom 𝐹 = dom (𝐴 × 𝐴)) | |
3 | dmxpid 5799 | . . 3 ⊢ dom (𝐴 × 𝐴) = 𝐴 | |
4 | 2, 3 | eqtrdi 2794 | . 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 1543 × cxp 5549 dom cdm 5551 Fn wfn 6375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-br 5054 df-opab 5116 df-xp 5557 df-dm 5561 df-fn 6383 |
This theorem is referenced by: (None) |
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