Theorem imaundir 5041

Description: The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.)

Assertion

Ref	Expression
imaundir	⊢ ((A ∪ B) “ C) = ((A “ C) ∪ (B “ C))

Proof of Theorem imaundir

Step	Hyp	Ref	Expression
1		dfima3 4952	. . 3 ⊢ ((A ∪ B) “ C) = ran ((A ∪ B) ↾ C)
2		resundir 4983	. . . 4 ⊢ ((A ∪ B) ↾ C) = ((A ↾ C) ∪ (B ↾ C))
3	2	rneqi 4958	. . 3 ⊢ ran ((A ∪ B) ↾ C) = ran ((A ↾ C) ∪ (B ↾ C))
4		rnun 5037	. . 3 ⊢ ran ((A ↾ C) ∪ (B ↾ C)) = (ran (A ↾ C) ∪ ran (B ↾ C))
5	1, 3, 4	3eqtri 2377	. 2 ⊢ ((A ∪ B) “ C) = (ran (A ↾ C) ∪ ran (B ↾ C))
6		dfima3 4952	. . 3 ⊢ (A “ C) = ran (A ↾ C)
7		dfima3 4952	. . 3 ⊢ (B “ C) = ran (B ↾ C)
8	6, 7	uneq12i 3417	. 2 ⊢ ((A “ C) ∪ (B “ C)) = (ran (A ↾ C) ∪ ran (B ↾ C))
9	5, 8	eqtr4i 2376	1 ⊢ ((A ∪ B) “ C) = ((A “ C) ∪ (B “ C))

Syntax hints:   = wceq 1642   ∪ cun 3208   “ cima 4723  ran crn 4774   ↾ cres 4775

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-ima 4728  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789

This theorem is referenced by:  fvun  5379