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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 6448 | . 2 ⊢ (𝐹 Fn (𝐴 × 𝐴) → dom 𝐹 = (𝐴 × 𝐴)) | |
2 | dmeq 5765 | . . 3 ⊢ (dom 𝐹 = (𝐴 × 𝐴) → dom dom 𝐹 = dom (𝐴 × 𝐴)) | |
3 | dmxpid 5793 | . . 3 ⊢ dom (𝐴 × 𝐴) = 𝐴 | |
4 | 2, 3 | syl6eq 2869 | . 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 1528 × cxp 5546 dom cdm 5548 Fn wfn 6343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-dm 5558 df-fn 6351 |
This theorem is referenced by: (None) |
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