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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 5990 | . 2 ⊢ (𝐹 Fn (𝐴 × 𝐴) → dom 𝐹 = (𝐴 × 𝐴)) | |
2 | dmeq 5324 | . . 3 ⊢ (dom 𝐹 = (𝐴 × 𝐴) → dom dom 𝐹 = dom (𝐴 × 𝐴)) | |
3 | dmxpid 5345 | . . 3 ⊢ dom (𝐴 × 𝐴) = 𝐴 | |
4 | 2, 3 | syl6eq 2672 | . 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 1483 × cxp 5112 dom cdm 5114 Fn wfn 5883 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-dm 5124 df-fn 5891 |
This theorem is referenced by: (None) |
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