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Theorem rankung 35138
Description: The rank of the union of two sets. Closed form of rankun 9851. (Contributed by Scott Fenton, 15-Jul-2015.)
Assertion
Ref Expression
rankung ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (rankβ€˜(𝐴 βˆͺ 𝐡)) = ((rankβ€˜π΄) βˆͺ (rankβ€˜π΅)))

Proof of Theorem rankung
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uneq1 4157 . . . 4 (π‘₯ = 𝐴 β†’ (π‘₯ βˆͺ 𝑦) = (𝐴 βˆͺ 𝑦))
21fveq2d 6896 . . 3 (π‘₯ = 𝐴 β†’ (rankβ€˜(π‘₯ βˆͺ 𝑦)) = (rankβ€˜(𝐴 βˆͺ 𝑦)))
3 fveq2 6892 . . . 4 (π‘₯ = 𝐴 β†’ (rankβ€˜π‘₯) = (rankβ€˜π΄))
43uneq1d 4163 . . 3 (π‘₯ = 𝐴 β†’ ((rankβ€˜π‘₯) βˆͺ (rankβ€˜π‘¦)) = ((rankβ€˜π΄) βˆͺ (rankβ€˜π‘¦)))
52, 4eqeq12d 2749 . 2 (π‘₯ = 𝐴 β†’ ((rankβ€˜(π‘₯ βˆͺ 𝑦)) = ((rankβ€˜π‘₯) βˆͺ (rankβ€˜π‘¦)) ↔ (rankβ€˜(𝐴 βˆͺ 𝑦)) = ((rankβ€˜π΄) βˆͺ (rankβ€˜π‘¦))))
6 uneq2 4158 . . . 4 (𝑦 = 𝐡 β†’ (𝐴 βˆͺ 𝑦) = (𝐴 βˆͺ 𝐡))
76fveq2d 6896 . . 3 (𝑦 = 𝐡 β†’ (rankβ€˜(𝐴 βˆͺ 𝑦)) = (rankβ€˜(𝐴 βˆͺ 𝐡)))
8 fveq2 6892 . . . 4 (𝑦 = 𝐡 β†’ (rankβ€˜π‘¦) = (rankβ€˜π΅))
98uneq2d 4164 . . 3 (𝑦 = 𝐡 β†’ ((rankβ€˜π΄) βˆͺ (rankβ€˜π‘¦)) = ((rankβ€˜π΄) βˆͺ (rankβ€˜π΅)))
107, 9eqeq12d 2749 . 2 (𝑦 = 𝐡 β†’ ((rankβ€˜(𝐴 βˆͺ 𝑦)) = ((rankβ€˜π΄) βˆͺ (rankβ€˜π‘¦)) ↔ (rankβ€˜(𝐴 βˆͺ 𝐡)) = ((rankβ€˜π΄) βˆͺ (rankβ€˜π΅))))
11 vex 3479 . . 3 π‘₯ ∈ V
12 vex 3479 . . 3 𝑦 ∈ V
1311, 12rankun 9851 . 2 (rankβ€˜(π‘₯ βˆͺ 𝑦)) = ((rankβ€˜π‘₯) βˆͺ (rankβ€˜π‘¦))
145, 10, 13vtocl2g 3563 1 ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (rankβ€˜(𝐴 βˆͺ 𝐡)) = ((rankβ€˜π΄) βˆͺ (rankβ€˜π΅)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   βˆͺ cun 3947  β€˜cfv 6544  rankcrnk 9758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-reg 9587  ax-inf2 9636
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-r1 9759  df-rank 9760
This theorem is referenced by:  hfun  35150
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