Theorem imauni 5662

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 3866	. . 3 ⊢ ∪ 𝐵 = ∪ 𝑥 ∈ 𝐵 𝑥
2	1	imaeq2i 4879	. 2 ⊢ (𝐴 “ ∪ 𝐵) = (𝐴 “ ∪ 𝑥 ∈ 𝐵 𝑥)
3		imaiun 5661	. 2 ⊢ (𝐴 “ ∪ 𝑥 ∈ 𝐵 𝑥) = ∪ 𝑥 ∈ 𝐵 (𝐴 “ 𝑥)
4	2, 3	eqtri 2160	1 ⊢ (𝐴 “ ∪ 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 “ 𝑥)

Syntax hints:   = wceq 1331  ∪ cuni 3736  ∪ ciun 3813   “ cima 4542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-xp 4545  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552

This theorem is referenced by:  tgcn  12377