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Mirrors > Home > MPE Home > Th. List > ustuni | Structured version Visualization version GIF version |
Description: The set union of a uniform structure is the Cartesian product of its base. (Contributed by Thierry Arnoux, 5-Dec-2017.) |
Ref | Expression |
---|---|
ustuni | β’ (π β (UnifOnβπ) β βͺ π = (π Γ π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ustbasel 23574 | . 2 β’ (π β (UnifOnβπ) β (π Γ π) β π) | |
2 | ustssxp 23572 | . . . 4 β’ ((π β (UnifOnβπ) β§ π’ β π) β π’ β (π Γ π)) | |
3 | 2 | ralrimiva 3140 | . . 3 β’ (π β (UnifOnβπ) β βπ’ β π π’ β (π Γ π)) |
4 | pwssb 5062 | . . 3 β’ (π β π« (π Γ π) β βπ’ β π π’ β (π Γ π)) | |
5 | 3, 4 | sylibr 233 | . 2 β’ (π β (UnifOnβπ) β π β π« (π Γ π)) |
6 | elpwuni 5066 | . . 3 β’ ((π Γ π) β π β (π β π« (π Γ π) β βͺ π = (π Γ π))) | |
7 | 6 | biimpa 478 | . 2 β’ (((π Γ π) β π β§ π β π« (π Γ π)) β βͺ π = (π Γ π)) |
8 | 1, 5, 7 | syl2anc 585 | 1 β’ (π β (UnifOnβπ) β βͺ π = (π Γ π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βwral 3061 β wss 3911 π« cpw 4561 βͺ cuni 4866 Γ cxp 5632 βcfv 6497 UnifOncust 23567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-res 5646 df-iota 6449 df-fun 6499 df-fv 6505 df-ust 23568 |
This theorem is referenced by: tususs 23638 cnflduss 24736 |
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