Theorem rankung 32273

Description: The rank of the union of two sets. Closed form of rankun 8719. (Contributed by Scott Fenton, 15-Jul-2015.)

Assertion

Ref	Expression
rankung	|- ( ( A e. V /\ B e. W ) -> ( rank ` ( A u. B ) ) = ( ( rank ` A ) u. ( 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 3760	. . . 4 |- ( x = A -> ( x u. y ) = ( A u. y ) )
2	1	fveq2d 6195	. . 3 |- ( x = A -> ( rank ` ( x u. y ) ) = ( rank ` ( A u. y ) ) )
3		fveq2 6191	. . . 4 |- ( x = A -> ( rank ` x ) = ( rank ` A ) )
4	3	uneq1d 3766	. . 3 |- ( x = A -> ( ( rank ` x ) u. ( rank ` y ) ) = ( ( rank ` A ) u. ( rank ` y ) ) )
5	2, 4	eqeq12d 2637	. 2 |- ( x = A -> ( ( rank ` ( x u. y ) ) = ( ( rank ` x ) u. ( rank ` y ) ) <-> ( rank ` ( A u. y ) ) = ( ( rank ` A ) u. ( rank ` y ) ) ) )
6		uneq2 3761	. . . 4 |- ( y = B -> ( A u. y ) = ( A u. B ) )
7	6	fveq2d 6195	. . 3 |- ( y = B -> ( rank ` ( A u. y ) ) = ( rank ` ( A u. B ) ) )
8		fveq2 6191	. . . 4 |- ( y = B -> ( rank ` y ) = ( rank ` B ) )
9	8	uneq2d 3767	. . 3 |- ( y = B -> ( ( rank ` A ) u. ( rank ` y ) ) = ( ( rank ` A ) u. ( rank ` B ) ) )
10	7, 9	eqeq12d 2637	. 2 |- ( y = B -> ( ( rank ` ( A u. y ) ) = ( ( rank ` A ) u. ( rank ` y ) ) <-> ( rank ` ( A u. B ) ) = ( ( rank ` A ) u. ( rank ` B ) ) ) )
11		vex 3203	. . 3 |- x e. _V
12		vex 3203	. . 3 |- y e. _V
13	11, 12	rankun 8719	. 2 |- ( rank ` ( x u. y ) ) = ( ( rank ` x ) u. ( rank ` y ) )
14	5, 10, 13	vtocl2g 3270	1 |- ( ( A e. V /\ B e. W ) -> ( rank ` ( A u. B ) ) = ( ( rank ` A ) u. ( rank ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   u. cun 3572   ` cfv 5888   rankcrnk 8626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-r1 8627  df-rank 8628

This theorem is referenced by:  hfun  32285