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Theorem rnco 5087
 Description: The range of the composition of two classes. (Contributed by set.mm contributors, 12-Dec-2006.)
Assertion
Ref Expression
rnco ran (A B) = ran (A ran B)

Proof of Theorem rnco
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brco 4883 . . . . 5 (x(A B)yz(xBz zAy))
21exbii 1582 . . . 4 (x x(A B)yxz(xBz zAy))
3 excom 1741 . . . 4 (xz(xBz zAy) ↔ zx(xBz zAy))
4 ancom 437 . . . . . . 7 ((x xBz zAy) ↔ (zAy x xBz))
5 19.41v 1901 . . . . . . 7 (x(xBz zAy) ↔ (x xBz zAy))
6 elrn 4896 . . . . . . . 8 (z ran Bx xBz)
76anbi2i 675 . . . . . . 7 ((zAy z ran B) ↔ (zAy x xBz))
84, 5, 73bitr4i 268 . . . . . 6 (x(xBz zAy) ↔ (zAy z ran B))
9 brres 4949 . . . . . 6 (z(A ran B)y ↔ (zAy z ran B))
108, 9bitr4i 243 . . . . 5 (x(xBz zAy) ↔ z(A ran B)y)
1110exbii 1582 . . . 4 (zx(xBz zAy) ↔ z z(A ran B)y)
122, 3, 113bitri 262 . . 3 (x x(A B)yz z(A ran B)y)
13 elrn 4896 . . 3 (y ran (A B) ↔ x x(A B)y)
14 elrn 4896 . . 3 (y ran (A ran B) ↔ z z(A ran B)y)
1512, 13, 143bitr4i 268 . 2 (y ran (A B) ↔ y ran (A ran B))
1615eqriv 2350 1 ran (A B) = ran (A ran B)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   class class class wbr 4639   ∘ ccom 4721  ran crn 4773   ↾ cres 4774 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-xp 4784  df-rn 4786  df-res 4788 This theorem is referenced by:  rnco2  5088
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