Theorem imaiun 5465

Description: The image of an indexed union is the indexed union of the images. (Contributed by Mario Carneiro, 18-Jun-2014.)

Assertion

Ref	Expression
imaiun	⊢ (A “ ∪x ∈ B C) = ∪x ∈ B (A “ C)

Distinct variable group:   x,A

Allowed substitution hints:   B(x)   C(x)

Proof of Theorem imaiun

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexcom4 2879	. . . 4 ⊢ (∃x ∈ B ∃z(z ∈ C ∧ ⟨z, y⟩ ∈ A) ↔ ∃z∃x ∈ B (z ∈ C ∧ ⟨z, y⟩ ∈ A))
2		elima3 4757	. . . . 5 ⊢ (y ∈ (A “ C) ↔ ∃z(z ∈ C ∧ ⟨z, y⟩ ∈ A))
3	2	rexbii 2640	. . . 4 ⊢ (∃x ∈ B y ∈ (A “ C) ↔ ∃x ∈ B ∃z(z ∈ C ∧ ⟨z, y⟩ ∈ A))
4		eliun 3974	. . . . . . 7 ⊢ (z ∈ ∪x ∈ B C ↔ ∃x ∈ B z ∈ C)
5	4	anbi1i 676	. . . . . 6 ⊢ ((z ∈ ∪x ∈ B C ∧ ⟨z, y⟩ ∈ A) ↔ (∃x ∈ B z ∈ C ∧ ⟨z, y⟩ ∈ A))
6		r19.41v 2765	. . . . . 6 ⊢ (∃x ∈ B (z ∈ C ∧ ⟨z, y⟩ ∈ A) ↔ (∃x ∈ B z ∈ C ∧ ⟨z, y⟩ ∈ A))
7	5, 6	bitr4i 243	. . . . 5 ⊢ ((z ∈ ∪x ∈ B C ∧ ⟨z, y⟩ ∈ A) ↔ ∃x ∈ B (z ∈ C ∧ ⟨z, y⟩ ∈ A))
8	7	exbii 1582	. . . 4 ⊢ (∃z(z ∈ ∪x ∈ B C ∧ ⟨z, y⟩ ∈ A) ↔ ∃z∃x ∈ B (z ∈ C ∧ ⟨z, y⟩ ∈ A))
9	1, 3, 8	3bitr4ri 269	. . 3 ⊢ (∃z(z ∈ ∪x ∈ B C ∧ ⟨z, y⟩ ∈ A) ↔ ∃x ∈ B y ∈ (A “ C))
10		elima3 4757	. . 3 ⊢ (y ∈ (A “ ∪x ∈ B C) ↔ ∃z(z ∈ ∪x ∈ B C ∧ ⟨z, y⟩ ∈ A))
11		eliun 3974	. . 3 ⊢ (y ∈ ∪x ∈ B (A “ C) ↔ ∃x ∈ B y ∈ (A “ C))
12	9, 10, 11	3bitr4i 268	. 2 ⊢ (y ∈ (A “ ∪x ∈ B C) ↔ y ∈ ∪x ∈ B (A “ C))
13	12	eqriv 2350	1 ⊢ (A “ ∪x ∈ B C) = ∪x ∈ B (A “ C)

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  ∪ciun 3970  ⟨cop 4562   “ cima 4723

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-iun 3972  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-br 4641  df-ima 4728

This theorem is referenced by:  imauni  5466  uniqs  5985