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Theorem rankung 31443
Description: The rank of the union of two sets. Closed form of rankun 8602. (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 3722 . . . 4 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
21fveq2d 6107 . . 3 (𝑥 = 𝐴 → (rank‘(𝑥𝑦)) = (rank‘(𝐴𝑦)))
3 fveq2 6103 . . . 4 (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
43uneq1d 3728 . . 3 (𝑥 = 𝐴 → ((rank‘𝑥) ∪ (rank‘𝑦)) = ((rank‘𝐴) ∪ (rank‘𝑦)))
52, 4eqeq12d 2625 . 2 (𝑥 = 𝐴 → ((rank‘(𝑥𝑦)) = ((rank‘𝑥) ∪ (rank‘𝑦)) ↔ (rank‘(𝐴𝑦)) = ((rank‘𝐴) ∪ (rank‘𝑦))))
6 uneq2 3723 . . . 4 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
76fveq2d 6107 . . 3 (𝑦 = 𝐵 → (rank‘(𝐴𝑦)) = (rank‘(𝐴𝐵)))
8 fveq2 6103 . . . 4 (𝑦 = 𝐵 → (rank‘𝑦) = (rank‘𝐵))
98uneq2d 3729 . . 3 (𝑦 = 𝐵 → ((rank‘𝐴) ∪ (rank‘𝑦)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
107, 9eqeq12d 2625 . 2 (𝑦 = 𝐵 → ((rank‘(𝐴𝑦)) = ((rank‘𝐴) ∪ (rank‘𝑦)) ↔ (rank‘(𝐴𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))))
11 vex 3176 . . 3 𝑥 ∈ V
12 vex 3176 . . 3 𝑦 ∈ V
1311, 12rankun 8602 . 2 (rank‘(𝑥𝑦)) = ((rank‘𝑥) ∪ (rank‘𝑦))
145, 10, 13vtocl2g 3243 1 ((𝐴𝑉𝐵𝑊) → (rank‘(𝐴𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  cun 3538  cfv 5804  rankcrnk 8509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-inf2 8421
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-r1 8510  df-rank 8511
This theorem is referenced by:  hfun  31455
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