Theorem dfima2 5468

Description: Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (Contributed by NM, 19-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
dfima2	⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦}

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem dfima2

Step	Hyp	Ref	Expression
1		df-ima 5127	. 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
2		dfrn2 5311	. 2 ⊢ ran (𝐴 ↾ 𝐵) = {𝑦 ∣ ∃𝑥 𝑥(𝐴 ↾ 𝐵)𝑦}
3		vex 3203	. . . . . . 7 ⊢ 𝑦 ∈ V
4	3	brres 5402	. . . . . 6 ⊢ (𝑥(𝐴 ↾ 𝐵)𝑦 ↔ (𝑥𝐴𝑦 ∧ 𝑥 ∈ 𝐵))
5		ancom 466	. . . . . 6 ⊢ ((𝑥𝐴𝑦 ∧ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥𝐴𝑦))
6	4, 5	bitri 264	. . . . 5 ⊢ (𝑥(𝐴 ↾ 𝐵)𝑦 ↔ (𝑥 ∈ 𝐵 ∧ 𝑥𝐴𝑦))
7	6	exbii 1774	. . . 4 ⊢ (∃𝑥 𝑥(𝐴 ↾ 𝐵)𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐴𝑦))
8		df-rex 2918	. . . 4 ⊢ (∃𝑥 ∈ 𝐵 𝑥𝐴𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐴𝑦))
9	7, 8	bitr4i 267	. . 3 ⊢ (∃𝑥 𝑥(𝐴 ↾ 𝐵)𝑦 ↔ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦)
10	9	abbii 2739	. 2 ⊢ {𝑦 ∣ ∃𝑥 𝑥(𝐴 ↾ 𝐵)𝑦} = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦}
11	1, 2, 10	3eqtri 2648	1 ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦}

Syntax hints:   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608  ∃wrex 2913   class class class wbr 4653  ran crn 5115   ↾ cres 5116   “ cima 5117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  dfima3  5469  elimag  5470  imasng  5487  dfimafn  6245  isoini  6588  dffin1-5  9210  dfimafnf  29436  ofpreima  29465  dfaimafn  41245