Theorem dfima2 2604

Description: Alternate definition of image. Compare definition (d) of [Enderton] p. 44.

Assertion

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

Distinct variable group(s):   x,y,A   x,B,y

Proof of Theorem dfima2

Step	Hyp	Ref	Expression
1		df-ima 2431	. 2 ⊢ (A “ B) = ran (A ↾ B)
2		dfrn3 2524	. 2 ⊢ ran (A ↾ B) = {y∣∃x⟨x, y⟩ ∈ (A ↾ B)}
3		df-res 2430	. . . . . . 7 ⊢ (A ↾ B) = (A ∩ (B × V))
4	3	eleq2i 1153	. . . . . 6 ⊢ (⟨x, y⟩ ∈ (A ↾ B) ↔ ⟨x, y⟩ ∈ (A ∩ (B × V)))
5		elin 1635	. . . . . 6 ⊢ (⟨x, y⟩ ∈ (A ∩ (B × V)) ↔ (⟨x, y⟩ ∈ A ∧ ⟨x, y⟩ ∈ (B × V)))
6		ancom 333	. . . . . . 7 ⊢ ((x ∈ B ∧ xAy) ↔ (xAy ∧ x ∈ B))
7		df-br 2063	. . . . . . . 8 ⊢ (xAy ↔ ⟨x, y⟩ ∈ A)
8		visset 1350	. . . . . . . . . 10 ⊢ y ∈ V
9	8	biantru 543	. . . . . . . . 9 ⊢ (x ∈ B ↔ (x ∈ B ∧ y ∈ V))
10	8	opelxp 2452	. . . . . . . . 9 ⊢ (⟨x, y⟩ ∈ (B × V) ↔ (x ∈ B ∧ y ∈ V))
11	9, 10	bitr4 154	. . . . . . . 8 ⊢ (x ∈ B ↔ ⟨x, y⟩ ∈ (B × V))
12	7, 11	anbi12i 369	. . . . . . 7 ⊢ ((xAy ∧ x ∈ B) ↔ (⟨x, y⟩ ∈ A ∧ ⟨x, y⟩ ∈ (B × V)))
13	6, 12	bitr2 152	. . . . . 6 ⊢ ((⟨x, y⟩ ∈ A ∧ ⟨x, y⟩ ∈ (B × V)) ↔ (x ∈ B ∧ xAy))
14	4, 5, 13	3bitr 155	. . . . 5 ⊢ (⟨x, y⟩ ∈ (A ↾ B) ↔ (x ∈ B ∧ xAy))
15	14	biex 733	. . . 4 ⊢ (∃x⟨x, y⟩ ∈ (A ↾ B) ↔ ∃x(x ∈ B ∧ xAy))
16		df-rex 1206	. . . 4 ⊢ (∃x ∈ B xAy ↔ ∃x(x ∈ B ∧ xAy))
17	15, 16	bitr4 154	. . 3 ⊢ (∃x⟨x, y⟩ ∈ (A ↾ B) ↔ ∃x ∈ B xAy)
18	17	biabi 1181	. 2 ⊢ {y∣∃x⟨x, y⟩ ∈ (A ↾ B)} = {y∣∃x ∈ B xAy}
19	1, 2, 18	3eqtr 1123	1 ⊢ (A “ B) = {y∣∃x ∈ B xAy}

Syntax hints:   ∧ wa 196  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  Vcvv 1348   ∩ cin 1486  ⟨cop 1810   class class class wbr 2054   × cxp 2408  ran crn 2411   ↾ cres 2412   “ cima 2413

This theorem is referenced by:  dfima3 2605  elima 2606  fv2 2828  isoini 2938

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431

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