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Theorem rankung 32257
 Description: The rank of the union of two sets. Closed form of rankun 8716. (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 3758 . . . 4 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
21fveq2d 6193 . . 3 (𝑥 = 𝐴 → (rank‘(𝑥𝑦)) = (rank‘(𝐴𝑦)))
3 fveq2 6189 . . . 4 (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
43uneq1d 3764 . . 3 (𝑥 = 𝐴 → ((rank‘𝑥) ∪ (rank‘𝑦)) = ((rank‘𝐴) ∪ (rank‘𝑦)))
52, 4eqeq12d 2636 . 2 (𝑥 = 𝐴 → ((rank‘(𝑥𝑦)) = ((rank‘𝑥) ∪ (rank‘𝑦)) ↔ (rank‘(𝐴𝑦)) = ((rank‘𝐴) ∪ (rank‘𝑦))))
6 uneq2 3759 . . . 4 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
76fveq2d 6193 . . 3 (𝑦 = 𝐵 → (rank‘(𝐴𝑦)) = (rank‘(𝐴𝐵)))
8 fveq2 6189 . . . 4 (𝑦 = 𝐵 → (rank‘𝑦) = (rank‘𝐵))
98uneq2d 3765 . . 3 (𝑦 = 𝐵 → ((rank‘𝐴) ∪ (rank‘𝑦)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
107, 9eqeq12d 2636 . 2 (𝑦 = 𝐵 → ((rank‘(𝐴𝑦)) = ((rank‘𝐴) ∪ (rank‘𝑦)) ↔ (rank‘(𝐴𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))))
11 vex 3201 . . 3 𝑥 ∈ V
12 vex 3201 . . 3 𝑦 ∈ V
1311, 12rankun 8716 . 2 (rank‘(𝑥𝑦)) = ((rank‘𝑥) ∪ (rank‘𝑦))
145, 10, 13vtocl2g 3268 1 ((𝐴𝑉𝐵𝑊) → (rank‘(𝐴𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1482   ∈ wcel 1989   ∪ cun 3570  ‘cfv 5886  rankcrnk 8623 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-reg 8494  ax-inf2 8535 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-om 7063  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-r1 8624  df-rank 8625 This theorem is referenced by:  hfun  32269
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