Theorem imauni 7281

Description: The image of a union is the indexed union of the images. Theorem 3K(a) of [Enderton] p. 50. (Contributed by NM, 9-Aug-2004.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem imauni

Step	Hyp	Ref	Expression
1		uniiun 5022	. . 3 ⊢ ∪ 𝐵 = ∪ 𝑥 ∈ 𝐵 𝑥
2	1	imaeq2i 6068	. 2 ⊢ (𝐴 “ ∪ 𝐵) = (𝐴 “ ∪ 𝑥 ∈ 𝐵 𝑥)
3		imaiun 7280	. 2 ⊢ (𝐴 “ ∪ 𝑥 ∈ 𝐵 𝑥) = ∪ 𝑥 ∈ 𝐵 (𝐴 “ 𝑥)
4	2, 3	eqtri 2786	1 ⊢ (𝐴 “ ∪ 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 “ 𝑥)

This proof depends on syntax axioms:   = wceq 1570  ∪ cuni 4872  ∪ ciun 4956   “ cima 5661

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-11 2197  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-xp 5664  df-cnv 5666  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671

This theorem is used by:  enfin2i  10438  tgcn  23625  cncmp  23765  qtoptop2  24073  mbfimaopnlem  26031  fnpreimac  33362  tmachlem-franscan  48232