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Theorem rnuni 5038
 Description: The range of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by set.mm contributors, 17-Mar-2004.)
Assertion
Ref Expression
rnuni ran A = x A ran x
Distinct variable group:   x,A

Proof of Theorem rnuni
Dummy variables y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluni 3894 . . . . . 6 (y, z Ax(y, z x x A))
21exbii 1582 . . . . 5 (yy, z Ayx(y, z x x A))
3 excom 1741 . . . . 5 (yx(y, z x x A) ↔ xy(y, z x x A))
4 elrn2 4897 . . . . . . . 8 (z ran xyy, z x)
54anbi1i 676 . . . . . . 7 ((z ran x x A) ↔ (yy, z x x A))
6 ancom 437 . . . . . . 7 ((x A z ran x) ↔ (z ran x x A))
7 19.41v 1901 . . . . . . 7 (y(y, z x x A) ↔ (yy, z x x A))
85, 6, 73bitr4ri 269 . . . . . 6 (y(y, z x x A) ↔ (x A z ran x))
98exbii 1582 . . . . 5 (xy(y, z x x A) ↔ x(x A z ran x))
102, 3, 93bitri 262 . . . 4 (yy, z Ax(x A z ran x))
11 df-rex 2620 . . . 4 (x A z ran xx(x A z ran x))
1210, 11bitr4i 243 . . 3 (yy, z Ax A z ran x)
13 elrn2 4897 . . 3 (z ran Ayy, z A)
14 eliun 3973 . . 3 (z x A ran xx A z ran x)
1512, 13, 143bitr4i 268 . 2 (z ran Az x A ran x)
1615eqriv 2350 1 ran A = x A ran x
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  ∪cuni 3891  ∪ciun 3969  ⟨cop 4561  ran crn 4773 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-iun 3971  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-br 4640  df-ima 4727  df-rn 4786 This theorem is referenced by: (None)
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