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Theorem rankung 33701
Description: The rank of the union of two sets. Closed form of rankun 9273. (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 4107 . . . 4 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
21fveq2d 6656 . . 3 (𝑥 = 𝐴 → (rank‘(𝑥𝑦)) = (rank‘(𝐴𝑦)))
3 fveq2 6652 . . . 4 (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
43uneq1d 4113 . . 3 (𝑥 = 𝐴 → ((rank‘𝑥) ∪ (rank‘𝑦)) = ((rank‘𝐴) ∪ (rank‘𝑦)))
52, 4eqeq12d 2838 . 2 (𝑥 = 𝐴 → ((rank‘(𝑥𝑦)) = ((rank‘𝑥) ∪ (rank‘𝑦)) ↔ (rank‘(𝐴𝑦)) = ((rank‘𝐴) ∪ (rank‘𝑦))))
6 uneq2 4108 . . . 4 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
76fveq2d 6656 . . 3 (𝑦 = 𝐵 → (rank‘(𝐴𝑦)) = (rank‘(𝐴𝐵)))
8 fveq2 6652 . . . 4 (𝑦 = 𝐵 → (rank‘𝑦) = (rank‘𝐵))
98uneq2d 4114 . . 3 (𝑦 = 𝐵 → ((rank‘𝐴) ∪ (rank‘𝑦)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
107, 9eqeq12d 2838 . 2 (𝑦 = 𝐵 → ((rank‘(𝐴𝑦)) = ((rank‘𝐴) ∪ (rank‘𝑦)) ↔ (rank‘(𝐴𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))))
11 vex 3472 . . 3 𝑥 ∈ V
12 vex 3472 . . 3 𝑦 ∈ V
1311, 12rankun 9273 . 2 (rank‘(𝑥𝑦)) = ((rank‘𝑥) ∪ (rank‘𝑦))
145, 10, 13vtocl2g 3547 1 ((𝐴𝑉𝐵𝑊) → (rank‘(𝐴𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  cun 3906  cfv 6334  rankcrnk 9180
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-reg 9044  ax-inf2 9092
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-om 7566  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-r1 9181  df-rank 9182
This theorem is referenced by:  hfun  33713
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