Theorem imauni 7223

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 5025	. . 3 ⊢ ∪ 𝐵 = ∪ 𝑥 ∈ 𝐵 𝑥
2	1	imaeq2i 6032	. 2 ⊢ (𝐴 “ ∪ 𝐵) = (𝐴 “ ∪ 𝑥 ∈ 𝐵 𝑥)
3		imaiun 7222	. 2 ⊢ (𝐴 “ ∪ 𝑥 ∈ 𝐵 𝑥) = ∪ 𝑥 ∈ 𝐵 (𝐴 “ 𝑥)
4	2, 3	eqtri 2753	1 ⊢ (𝐴 “ ∪ 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 “ 𝑥)

Syntax hints:   = wceq 1540  ∪ cuni 4874  ∪ ciun 4958   “ cima 5644

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-xp 5647  df-cnv 5649  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654

This theorem is referenced by:  enfin2i  10281  tgcn  23146  cncmp  23286  qtoptop2  23593  mbfimaopnlem  25563  fnpreimac  32602