Theorem rankung 32862

Description: The rank of the union of two sets. Closed form of rankun 9016. (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 3983	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦))
2	1	fveq2d 6450	. . 3 ⊢ (𝑥 = 𝐴 → (rank‘(𝑥 ∪ 𝑦)) = (rank‘(𝐴 ∪ 𝑦)))
3		fveq2 6446	. . . 4 ⊢ (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
4	3	uneq1d 3989	. . 3 ⊢ (𝑥 = 𝐴 → ((rank‘𝑥) ∪ (rank‘𝑦)) = ((rank‘𝐴) ∪ (rank‘𝑦)))
5	2, 4	eqeq12d 2793	. 2 ⊢ (𝑥 = 𝐴 → ((rank‘(𝑥 ∪ 𝑦)) = ((rank‘𝑥) ∪ (rank‘𝑦)) ↔ (rank‘(𝐴 ∪ 𝑦)) = ((rank‘𝐴) ∪ (rank‘𝑦))))
6		uneq2 3984	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵))
7	6	fveq2d 6450	. . 3 ⊢ (𝑦 = 𝐵 → (rank‘(𝐴 ∪ 𝑦)) = (rank‘(𝐴 ∪ 𝐵)))
8		fveq2 6446	. . . 4 ⊢ (𝑦 = 𝐵 → (rank‘𝑦) = (rank‘𝐵))
9	8	uneq2d 3990	. . 3 ⊢ (𝑦 = 𝐵 → ((rank‘𝐴) ∪ (rank‘𝑦)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
10	7, 9	eqeq12d 2793	. 2 ⊢ (𝑦 = 𝐵 → ((rank‘(𝐴 ∪ 𝑦)) = ((rank‘𝐴) ∪ (rank‘𝑦)) ↔ (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))))
11		vex 3401	. . 3 ⊢ 𝑥 ∈ V
12		vex 3401	. . 3 ⊢ 𝑦 ∈ V
13	11, 12	rankun 9016	. 2 ⊢ (rank‘(𝑥 ∪ 𝑦)) = ((rank‘𝑥) ∪ (rank‘𝑦))
14	5, 10, 13	vtocl2g 3471	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107   ∪ cun 3790  ‘cfv 6135  rankcrnk 8923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-reg 8786  ax-inf2 8835

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-om 7344  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-r1 8924  df-rank 8925

This theorem is referenced by:  hfun  32874