Theorem rankung 34395

Description: The rank of the union of two sets. Closed form of rankun 9545. (Contributed by Scott Fenton, 15-Jul-2015.)

Assertion

Ref	Expression
rankung	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))

Proof of Theorem rankung

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uneq1 4086	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦))
2	1	fveq2d 6760	. . 3 ⊢ (𝑥 = 𝐴 → (rank‘(𝑥 ∪ 𝑦)) = (rank‘(𝐴 ∪ 𝑦)))
3		fveq2 6756	. . . 4 ⊢ (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
4	3	uneq1d 4092	. . 3 ⊢ (𝑥 = 𝐴 → ((rank‘𝑥) ∪ (rank‘𝑦)) = ((rank‘𝐴) ∪ (rank‘𝑦)))
5	2, 4	eqeq12d 2754	. 2 ⊢ (𝑥 = 𝐴 → ((rank‘(𝑥 ∪ 𝑦)) = ((rank‘𝑥) ∪ (rank‘𝑦)) ↔ (rank‘(𝐴 ∪ 𝑦)) = ((rank‘𝐴) ∪ (rank‘𝑦))))
6		uneq2 4087	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵))
7	6	fveq2d 6760	. . 3 ⊢ (𝑦 = 𝐵 → (rank‘(𝐴 ∪ 𝑦)) = (rank‘(𝐴 ∪ 𝐵)))
8		fveq2 6756	. . . 4 ⊢ (𝑦 = 𝐵 → (rank‘𝑦) = (rank‘𝐵))
9	8	uneq2d 4093	. . 3 ⊢ (𝑦 = 𝐵 → ((rank‘𝐴) ∪ (rank‘𝑦)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
10	7, 9	eqeq12d 2754	. 2 ⊢ (𝑦 = 𝐵 → ((rank‘(𝐴 ∪ 𝑦)) = ((rank‘𝐴) ∪ (rank‘𝑦)) ↔ (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))))
11		vex 3426	. . 3 ⊢ 𝑥 ∈ V
12		vex 3426	. . 3 ⊢ 𝑦 ∈ V
13	11, 12	rankun 9545	. 2 ⊢ (rank‘(𝑥 ∪ 𝑦)) = ((rank‘𝑥) ∪ (rank‘𝑦))
14	5, 10, 13	vtocl2g 3500	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ∪ cun 3881  ‘cfv 6418  rankcrnk 9452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-reg 9281  ax-inf2 9329

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-r1 9453  df-rank 9454

This theorem is referenced by:  hfun  34407