Theorem funimaex 2716

Description: The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 1075. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29.

Hypothesis

Ref	Expression
zfrep5.1	⊢ B ∈ V

Assertion

Ref	Expression
funimaex	⊢ (Fun A → (A “ B) ∈ V)

Proof of Theorem funimaex

Step	Hyp	Ref	Expression
1		zfrep5.1	. 2 ⊢ B ∈ V
2		funimaexg 2715	. 2 ⊢ (B ∈ V → (Fun A → (A “ B) ∈ V))
3	1, 2	ax-mp 6	1 ⊢ (Fun A → (A “ B) ∈ V)

Syntax hints:   → wi 2   ∈ wcel 1092  Vcvv 1348   “ cima 2413  Fun wfun 2416

This theorem is referenced by:  isofrlem 2939  f1oweOLD 2944  tz7.44-3 2968  tz9.12lem2 3504  zornlem7 3609

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-eu 1009  df-mo 1010  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-id 2125  df-xp 2424  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432

metamath.org