Theorem imassrn 2611

Description: The image of a class is a subset of its range. Theorem 3.16(xi) of [Monk1] p. 39.

Assertion

Ref	Expression
imassrn	⊢ (A “ B) ⊆ ran A

Proof of Theorem imassrn

Step	Hyp	Ref	Expression
1		pm3.27 260	. . . 4 ⊢ ((x ∈ B ∧ ⟨x, y⟩ ∈ A) → ⟨x, y⟩ ∈ A)
2	1	19.22i 723	. . 3 ⊢ (∃x(x ∈ B ∧ ⟨x, y⟩ ∈ A) → ∃x⟨x, y⟩ ∈ A)
3	2	ss2abi 1552	. 2 ⊢ {y∣∃x(x ∈ B ∧ ⟨x, y⟩ ∈ A)} ⊆ {y∣∃x⟨x, y⟩ ∈ A}
4		dfima3 2605	. 2 ⊢ (A “ B) = {y∣∃x(x ∈ B ∧ ⟨x, y⟩ ∈ A)}
5		dfrn3 2524	. 2 ⊢ ran A = {y∣∃x⟨x, y⟩ ∈ A}
6	3, 4, 5	3sstr4 1539	1 ⊢ (A “ B) ⊆ ran A

Syntax hints:   ∧ wa 196  ∃wex 678  {cab 1090   ∈ wcel 1092   ⊆ wss 1487  ⟨cop 1810  ran crn 2411   “ cima 2413

This theorem is referenced by:  imaexg 2612  isofrlem 2939  f1oweOLD 2944  pw2en 3348  sbthlem2 3350  sbthlem3 3351  sbthlem5 3353  sbthlem6 3354  ssenen 3399  ssfi 3430  fiint 3445

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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