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Theorem rankung 32610
 Description: The rank of the union of two sets. Closed form of rankun 8887. (Contributed by Scott Fenton, 15-Jul-2015.)
Assertion
Ref Expression
rankung ((𝐴𝑉𝐵𝑊) → (rank‘(𝐴𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))

Proof of Theorem rankung
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uneq1 3911 . . . 4 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
21fveq2d 6337 . . 3 (𝑥 = 𝐴 → (rank‘(𝑥𝑦)) = (rank‘(𝐴𝑦)))
3 fveq2 6333 . . . 4 (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
43uneq1d 3917 . . 3 (𝑥 = 𝐴 → ((rank‘𝑥) ∪ (rank‘𝑦)) = ((rank‘𝐴) ∪ (rank‘𝑦)))
52, 4eqeq12d 2786 . 2 (𝑥 = 𝐴 → ((rank‘(𝑥𝑦)) = ((rank‘𝑥) ∪ (rank‘𝑦)) ↔ (rank‘(𝐴𝑦)) = ((rank‘𝐴) ∪ (rank‘𝑦))))
6 uneq2 3912 . . . 4 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
76fveq2d 6337 . . 3 (𝑦 = 𝐵 → (rank‘(𝐴𝑦)) = (rank‘(𝐴𝐵)))
8 fveq2 6333 . . . 4 (𝑦 = 𝐵 → (rank‘𝑦) = (rank‘𝐵))
98uneq2d 3918 . . 3 (𝑦 = 𝐵 → ((rank‘𝐴) ∪ (rank‘𝑦)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
107, 9eqeq12d 2786 . 2 (𝑦 = 𝐵 → ((rank‘(𝐴𝑦)) = ((rank‘𝐴) ∪ (rank‘𝑦)) ↔ (rank‘(𝐴𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))))
11 vex 3354 . . 3 𝑥 ∈ V
12 vex 3354 . . 3 𝑦 ∈ V
1311, 12rankun 8887 . 2 (rank‘(𝑥𝑦)) = ((rank‘𝑥) ∪ (rank‘𝑦))
145, 10, 13vtocl2g 3421 1 ((𝐴𝑉𝐵𝑊) → (rank‘(𝐴𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145   ∪ cun 3721  ‘cfv 6030  rankcrnk 8794 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-reg 8657  ax-inf2 8706 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-om 7217  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-r1 8795  df-rank 8796 This theorem is referenced by:  hfun  32622
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