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Mathbox for Scott Fenton |
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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 9016. (Contributed by Scott Fenton, 15-Jul-2015.) |
Ref | Expression |
---|---|
rankung | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uneq1 3983 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦)) | |
2 | 1 | fveq2d 6450 | . . 3 ⊢ (𝑥 = 𝐴 → (rank‘(𝑥 ∪ 𝑦)) = (rank‘(𝐴 ∪ 𝑦))) |
3 | fveq2 6446 | . . . 4 ⊢ (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴)) | |
4 | 3 | uneq1d 3989 | . . 3 ⊢ (𝑥 = 𝐴 → ((rank‘𝑥) ∪ (rank‘𝑦)) = ((rank‘𝐴) ∪ (rank‘𝑦))) |
5 | 2, 4 | eqeq12d 2793 | . 2 ⊢ (𝑥 = 𝐴 → ((rank‘(𝑥 ∪ 𝑦)) = ((rank‘𝑥) ∪ (rank‘𝑦)) ↔ (rank‘(𝐴 ∪ 𝑦)) = ((rank‘𝐴) ∪ (rank‘𝑦)))) |
6 | uneq2 3984 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵)) | |
7 | 6 | fveq2d 6450 | . . 3 ⊢ (𝑦 = 𝐵 → (rank‘(𝐴 ∪ 𝑦)) = (rank‘(𝐴 ∪ 𝐵))) |
8 | fveq2 6446 | . . . 4 ⊢ (𝑦 = 𝐵 → (rank‘𝑦) = (rank‘𝐵)) | |
9 | 8 | uneq2d 3990 | . . 3 ⊢ (𝑦 = 𝐵 → ((rank‘𝐴) ∪ (rank‘𝑦)) = ((rank‘𝐴) ∪ (rank‘𝐵))) |
10 | 7, 9 | eqeq12d 2793 | . 2 ⊢ (𝑦 = 𝐵 → ((rank‘(𝐴 ∪ 𝑦)) = ((rank‘𝐴) ∪ (rank‘𝑦)) ↔ (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))) |
11 | vex 3401 | . . 3 ⊢ 𝑥 ∈ V | |
12 | vex 3401 | . . 3 ⊢ 𝑦 ∈ V | |
13 | 11, 12 | rankun 9016 | . 2 ⊢ (rank‘(𝑥 ∪ 𝑦)) = ((rank‘𝑥) ∪ (rank‘𝑦)) |
14 | 5, 10, 13 | vtocl2g 3471 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∪ cun 3790 ‘cfv 6135 rankcrnk 8923 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-reg 8786 ax-inf2 8835 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-r1 8924 df-rank 8925 |
This theorem is referenced by: hfun 32874 |
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