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Mirrors > Home > MPE Home > Th. List > Mathboxes > rankung | Structured version Visualization version GIF version |
Description: The rank of the union of two sets. Closed form of rankun 9873. (Contributed by Scott Fenton, 15-Jul-2015.) |
Ref | Expression |
---|---|
rankung | β’ ((π΄ β π β§ π΅ β π) β (rankβ(π΄ βͺ π΅)) = ((rankβπ΄) βͺ (rankβπ΅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uneq1 4152 | . . . 4 β’ (π₯ = π΄ β (π₯ βͺ π¦) = (π΄ βͺ π¦)) | |
2 | 1 | fveq2d 6895 | . . 3 β’ (π₯ = π΄ β (rankβ(π₯ βͺ π¦)) = (rankβ(π΄ βͺ π¦))) |
3 | fveq2 6891 | . . . 4 β’ (π₯ = π΄ β (rankβπ₯) = (rankβπ΄)) | |
4 | 3 | uneq1d 4158 | . . 3 β’ (π₯ = π΄ β ((rankβπ₯) βͺ (rankβπ¦)) = ((rankβπ΄) βͺ (rankβπ¦))) |
5 | 2, 4 | eqeq12d 2744 | . 2 β’ (π₯ = π΄ β ((rankβ(π₯ βͺ π¦)) = ((rankβπ₯) βͺ (rankβπ¦)) β (rankβ(π΄ βͺ π¦)) = ((rankβπ΄) βͺ (rankβπ¦)))) |
6 | uneq2 4153 | . . . 4 β’ (π¦ = π΅ β (π΄ βͺ π¦) = (π΄ βͺ π΅)) | |
7 | 6 | fveq2d 6895 | . . 3 β’ (π¦ = π΅ β (rankβ(π΄ βͺ π¦)) = (rankβ(π΄ βͺ π΅))) |
8 | fveq2 6891 | . . . 4 β’ (π¦ = π΅ β (rankβπ¦) = (rankβπ΅)) | |
9 | 8 | uneq2d 4159 | . . 3 β’ (π¦ = π΅ β ((rankβπ΄) βͺ (rankβπ¦)) = ((rankβπ΄) βͺ (rankβπ΅))) |
10 | 7, 9 | eqeq12d 2744 | . 2 β’ (π¦ = π΅ β ((rankβ(π΄ βͺ π¦)) = ((rankβπ΄) βͺ (rankβπ¦)) β (rankβ(π΄ βͺ π΅)) = ((rankβπ΄) βͺ (rankβπ΅)))) |
11 | vex 3474 | . . 3 β’ π₯ β V | |
12 | vex 3474 | . . 3 β’ π¦ β V | |
13 | 11, 12 | rankun 9873 | . 2 β’ (rankβ(π₯ βͺ π¦)) = ((rankβπ₯) βͺ (rankβπ¦)) |
14 | 5, 10, 13 | vtocl2g 3559 | 1 β’ ((π΄ β π β§ π΅ β π) β (rankβ(π΄ βͺ π΅)) = ((rankβπ΄) βͺ (rankβπ΅))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 βͺ cun 3943 βcfv 6542 rankcrnk 9780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-reg 9609 ax-inf2 9658 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-r1 9781 df-rank 9782 |
This theorem is referenced by: hfun 35768 |
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