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Mirrors > Home > MPE Home > Th. List > joinval2lem | Structured version Visualization version GIF version |
Description: Lemma for joinval2 18334 and joineu 18335. (Contributed by NM, 12-Sep-2018.) TODO: combine this through joineu 18335 into joinlem 18336? |
Ref | Expression |
---|---|
joinval2.b | β’ π΅ = (BaseβπΎ) |
joinval2.l | β’ β€ = (leβπΎ) |
joinval2.j | β’ β¨ = (joinβπΎ) |
joinval2.k | β’ (π β πΎ β π) |
joinval2.x | β’ (π β π β π΅) |
joinval2.y | β’ (π β π β π΅) |
Ref | Expression |
---|---|
joinval2lem | β’ ((π β π΅ β§ π β π΅) β ((βπ¦ β {π, π}π¦ β€ π₯ β§ βπ§ β π΅ (βπ¦ β {π, π}π¦ β€ π§ β π₯ β€ π§)) β ((π β€ π₯ β§ π β€ π₯) β§ βπ§ β π΅ ((π β€ π§ β§ π β€ π§) β π₯ β€ π§)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 5152 | . . 3 β’ (π¦ = π β (π¦ β€ π₯ β π β€ π₯)) | |
2 | breq1 5152 | . . 3 β’ (π¦ = π β (π¦ β€ π₯ β π β€ π₯)) | |
3 | 1, 2 | ralprg 4699 | . 2 β’ ((π β π΅ β§ π β π΅) β (βπ¦ β {π, π}π¦ β€ π₯ β (π β€ π₯ β§ π β€ π₯))) |
4 | breq1 5152 | . . . . 5 β’ (π¦ = π β (π¦ β€ π§ β π β€ π§)) | |
5 | breq1 5152 | . . . . 5 β’ (π¦ = π β (π¦ β€ π§ β π β€ π§)) | |
6 | 4, 5 | ralprg 4699 | . . . 4 β’ ((π β π΅ β§ π β π΅) β (βπ¦ β {π, π}π¦ β€ π§ β (π β€ π§ β§ π β€ π§))) |
7 | 6 | imbi1d 342 | . . 3 β’ ((π β π΅ β§ π β π΅) β ((βπ¦ β {π, π}π¦ β€ π§ β π₯ β€ π§) β ((π β€ π§ β§ π β€ π§) β π₯ β€ π§))) |
8 | 7 | ralbidv 3178 | . 2 β’ ((π β π΅ β§ π β π΅) β (βπ§ β π΅ (βπ¦ β {π, π}π¦ β€ π§ β π₯ β€ π§) β βπ§ β π΅ ((π β€ π§ β§ π β€ π§) β π₯ β€ π§))) |
9 | 3, 8 | anbi12d 632 | 1 β’ ((π β π΅ β§ π β π΅) β ((βπ¦ β {π, π}π¦ β€ π₯ β§ βπ§ β π΅ (βπ¦ β {π, π}π¦ β€ π§ β π₯ β€ π§)) β ((π β€ π₯ β§ π β€ π₯) β§ βπ§ β π΅ ((π β€ π§ β§ π β€ π§) β π₯ β€ π§)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3062 {cpr 4631 class class class wbr 5149 βcfv 6544 Basecbs 17144 lecple 17204 joincjn 18264 |
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-ext 2704 |
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-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 |
This theorem is referenced by: joinval2 18334 joineu 18335 joindm3 47602 |
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