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Theorem rankung 35756
Description: The rank of the union of two sets. Closed form of rankun 9873. (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 4152 . . . 4 (π‘₯ = 𝐴 β†’ (π‘₯ βˆͺ 𝑦) = (𝐴 βˆͺ 𝑦))
21fveq2d 6895 . . 3 (π‘₯ = 𝐴 β†’ (rankβ€˜(π‘₯ βˆͺ 𝑦)) = (rankβ€˜(𝐴 βˆͺ 𝑦)))
3 fveq2 6891 . . . 4 (π‘₯ = 𝐴 β†’ (rankβ€˜π‘₯) = (rankβ€˜π΄))
43uneq1d 4158 . . 3 (π‘₯ = 𝐴 β†’ ((rankβ€˜π‘₯) βˆͺ (rankβ€˜π‘¦)) = ((rankβ€˜π΄) βˆͺ (rankβ€˜π‘¦)))
52, 4eqeq12d 2744 . 2 (π‘₯ = 𝐴 β†’ ((rankβ€˜(π‘₯ βˆͺ 𝑦)) = ((rankβ€˜π‘₯) βˆͺ (rankβ€˜π‘¦)) ↔ (rankβ€˜(𝐴 βˆͺ 𝑦)) = ((rankβ€˜π΄) βˆͺ (rankβ€˜π‘¦))))
6 uneq2 4153 . . . 4 (𝑦 = 𝐡 β†’ (𝐴 βˆͺ 𝑦) = (𝐴 βˆͺ 𝐡))
76fveq2d 6895 . . 3 (𝑦 = 𝐡 β†’ (rankβ€˜(𝐴 βˆͺ 𝑦)) = (rankβ€˜(𝐴 βˆͺ 𝐡)))
8 fveq2 6891 . . . 4 (𝑦 = 𝐡 β†’ (rankβ€˜π‘¦) = (rankβ€˜π΅))
98uneq2d 4159 . . 3 (𝑦 = 𝐡 β†’ ((rankβ€˜π΄) βˆͺ (rankβ€˜π‘¦)) = ((rankβ€˜π΄) βˆͺ (rankβ€˜π΅)))
107, 9eqeq12d 2744 . 2 (𝑦 = 𝐡 β†’ ((rankβ€˜(𝐴 βˆͺ 𝑦)) = ((rankβ€˜π΄) βˆͺ (rankβ€˜π‘¦)) ↔ (rankβ€˜(𝐴 βˆͺ 𝐡)) = ((rankβ€˜π΄) βˆͺ (rankβ€˜π΅))))
11 vex 3474 . . 3 π‘₯ ∈ V
12 vex 3474 . . 3 𝑦 ∈ V
1311, 12rankun 9873 . 2 (rankβ€˜(π‘₯ βˆͺ 𝑦)) = ((rankβ€˜π‘₯) βˆͺ (rankβ€˜π‘¦))
145, 10, 13vtocl2g 3559 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 395   = wceq 1534   ∈ wcel 2099   βˆͺ cun 3943  β€˜cfv 6542  rankcrnk 9780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-reg 9609  ax-inf2 9658
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-om 7865  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-r1 9781  df-rank 9782
This theorem is referenced by:  hfun  35768
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