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Mirrors > Home > MPE Home > Th. List > joinval2lem | Structured version Visualization version GIF version |
Description: Lemma for joinval2 18333 and joineu 18334. (Contributed by NM, 12-Sep-2018.) TODO: combine this through joineu 18334 into joinlem 18335? |
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 5151 | . . 3 β’ (π¦ = π β (π¦ β€ π₯ β π β€ π₯)) | |
2 | breq1 5151 | . . 3 β’ (π¦ = π β (π¦ β€ π₯ β π β€ π₯)) | |
3 | 1, 2 | ralprg 4698 | . 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 5151 | . . . . 5 β’ (π¦ = π β (π¦ β€ π§ β π β€ π§)) | |
5 | breq1 5151 | . . . . 5 β’ (π¦ = π β (π¦ β€ π§ β π β€ π§)) | |
6 | 4, 5 | ralprg 4698 | . . . 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 341 | . . 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 3177 | . 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 631 | 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 396 = wceq 1541 β wcel 2106 βwral 3061 {cpr 4630 class class class wbr 5148 βcfv 6543 Basecbs 17143 lecple 17203 joincjn 18263 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 |
This theorem is referenced by: joinval2 18333 joineu 18334 joindm3 47592 |
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