Theorem rankung 36341

Description: The rank of the union of two sets. Closed form of rankun 9772. (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 4114	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦))
2	1	fveq2d 6839	. . 3 ⊢ (𝑥 = 𝐴 → (rank‘(𝑥 ∪ 𝑦)) = (rank‘(𝐴 ∪ 𝑦)))
3		fveq2 6835	. . . 4 ⊢ (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
4	3	uneq1d 4120	. . 3 ⊢ (𝑥 = 𝐴 → ((rank‘𝑥) ∪ (rank‘𝑦)) = ((rank‘𝐴) ∪ (rank‘𝑦)))
5	2, 4	eqeq12d 2753	. 2 ⊢ (𝑥 = 𝐴 → ((rank‘(𝑥 ∪ 𝑦)) = ((rank‘𝑥) ∪ (rank‘𝑦)) ↔ (rank‘(𝐴 ∪ 𝑦)) = ((rank‘𝐴) ∪ (rank‘𝑦))))
6		uneq2 4115	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵))
7	6	fveq2d 6839	. . 3 ⊢ (𝑦 = 𝐵 → (rank‘(𝐴 ∪ 𝑦)) = (rank‘(𝐴 ∪ 𝐵)))
8		fveq2 6835	. . . 4 ⊢ (𝑦 = 𝐵 → (rank‘𝑦) = (rank‘𝐵))
9	8	uneq2d 4121	. . 3 ⊢ (𝑦 = 𝐵 → ((rank‘𝐴) ∪ (rank‘𝑦)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
10	7, 9	eqeq12d 2753	. 2 ⊢ (𝑦 = 𝐵 → ((rank‘(𝐴 ∪ 𝑦)) = ((rank‘𝐴) ∪ (rank‘𝑦)) ↔ (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))))
11		vex 3445	. . 3 ⊢ 𝑥 ∈ V
12		vex 3445	. . 3 ⊢ 𝑦 ∈ V
13	11, 12	rankun 9772	. 2 ⊢ (rank‘(𝑥 ∪ 𝑦)) = ((rank‘𝑥) ∪ (rank‘𝑦))
14	5, 10, 13	vtocl2g 3530	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ∪ cun 3900  ‘cfv 6493  rankcrnk 9679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-reg 9501  ax-inf2 9554

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-om 7811  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-r1 9680  df-rank 9681

This theorem is referenced by:  hfun  36353