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Theorem rankung 31278
Description: The rank of the union of two sets. Closed form of rankun 8478. (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 3626 . . . 4 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
21fveq2d 5991 . . 3 (𝑥 = 𝐴 → (rank‘(𝑥𝑦)) = (rank‘(𝐴𝑦)))
3 fveq2 5987 . . . 4 (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
43uneq1d 3632 . . 3 (𝑥 = 𝐴 → ((rank‘𝑥) ∪ (rank‘𝑦)) = ((rank‘𝐴) ∪ (rank‘𝑦)))
52, 4eqeq12d 2529 . 2 (𝑥 = 𝐴 → ((rank‘(𝑥𝑦)) = ((rank‘𝑥) ∪ (rank‘𝑦)) ↔ (rank‘(𝐴𝑦)) = ((rank‘𝐴) ∪ (rank‘𝑦))))
6 uneq2 3627 . . . 4 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
76fveq2d 5991 . . 3 (𝑦 = 𝐵 → (rank‘(𝐴𝑦)) = (rank‘(𝐴𝐵)))
8 fveq2 5987 . . . 4 (𝑦 = 𝐵 → (rank‘𝑦) = (rank‘𝐵))
98uneq2d 3633 . . 3 (𝑦 = 𝐵 → ((rank‘𝐴) ∪ (rank‘𝑦)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
107, 9eqeq12d 2529 . 2 (𝑦 = 𝐵 → ((rank‘(𝐴𝑦)) = ((rank‘𝐴) ∪ (rank‘𝑦)) ↔ (rank‘(𝐴𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))))
11 vex 3080 . . 3 𝑥 ∈ V
12 vex 3080 . . 3 𝑦 ∈ V
1311, 12rankun 8478 . 2 (rank‘(𝑥𝑦)) = ((rank‘𝑥) ∪ (rank‘𝑦))
145, 10, 13vtocl2g 3147 1 ((𝐴𝑉𝐵𝑊) → (rank‘(𝐴𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1938  cun 3442  cfv 5689  rankcrnk 8385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6723  ax-reg 8256  ax-inf2 8297
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-int 4309  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-om 6834  df-wrecs 7169  df-recs 7231  df-rdg 7269  df-r1 8386  df-rank 8387
This theorem is referenced by:  hfun  31290
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