Theorem rankung 9900

Description: The rank of the union of two sets. Closed form of rankun 9897. (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 4163	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦))
2	1	fveq2d 6912	. . 3 ⊢ (𝑥 = 𝐴 → (rank‘(𝑥 ∪ 𝑦)) = (rank‘(𝐴 ∪ 𝑦)))
3		fveq2 6908	. . . 4 ⊢ (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
4	3	uneq1d 4169	. . 3 ⊢ (𝑥 = 𝐴 → ((rank‘𝑥) ∪ (rank‘𝑦)) = ((rank‘𝐴) ∪ (rank‘𝑦)))
5	2, 4	eqeq12d 2779	. 2 ⊢ (𝑥 = 𝐴 → ((rank‘(𝑥 ∪ 𝑦)) = ((rank‘𝑥) ∪ (rank‘𝑦)) ↔ (rank‘(𝐴 ∪ 𝑦)) = ((rank‘𝐴) ∪ (rank‘𝑦))))
6		uneq2 4164	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵))
7	6	fveq2d 6912	. . 3 ⊢ (𝑦 = 𝐵 → (rank‘(𝐴 ∪ 𝑦)) = (rank‘(𝐴 ∪ 𝐵)))
8		fveq2 6908	. . . 4 ⊢ (𝑦 = 𝐵 → (rank‘𝑦) = (rank‘𝐵))
9	8	uneq2d 4170	. . 3 ⊢ (𝑦 = 𝐵 → ((rank‘𝐴) ∪ (rank‘𝑦)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
10	7, 9	eqeq12d 2779	. 2 ⊢ (𝑦 = 𝐵 → ((rank‘(𝐴 ∪ 𝑦)) = ((rank‘𝐴) ∪ (rank‘𝑦)) ↔ (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))))
11		vex 3458	. . 3 ⊢ 𝑥 ∈ V
12		vex 3458	. . 3 ⊢ 𝑦 ∈ V
13	11, 12	rankun 9897	. 2 ⊢ (rank‘(𝑥 ∪ 𝑦)) = ((rank‘𝑥) ∪ (rank‘𝑦))
14	5, 10, 13	vtocl2g 3537	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ∪ cun 4152  ‘cfv 6558  rankcrnk 9794

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-reg 9613  ax-inf2 9669

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-om 7902  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-r1 9795  df-rank 9796

This theorem is used by:  hfunOLD  9946