Theorem imaeq1 4938

Description: Equality theorem for image. (Contributed by set.mm contributors, 14-Aug-1994.)

Assertion

Ref	Expression
imaeq1	⊢ (A = B → (A “ C) = (B “ C))

Proof of Theorem imaeq1

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq 4642	. . . 4 ⊢ (A = B → (yAx ↔ yBx))
2	1	rexbidv 2636	. . 3 ⊢ (A = B → (∃y ∈ C yAx ↔ ∃y ∈ C yBx))
3	2	abbidv 2468	. 2 ⊢ (A = B → {x ∣ ∃y ∈ C yAx} = {x ∣ ∃y ∈ C yBx})
4		df-ima 4728	. 2 ⊢ (A “ C) = {x ∣ ∃y ∈ C yAx}
5		df-ima 4728	. 2 ⊢ (B “ C) = {x ∣ ∃y ∈ C yBx}
6	3, 4, 5	3eqtr4g 2410	1 ⊢ (A = B → (A “ C) = (B “ C))

Syntax hints:   → wi 4   = wceq 1642  {cab 2339  ∃wrex 2616   class class class wbr 4640   “ cima 4723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-rex 2621  df-br 4641  df-ima 4728

This theorem is referenced by:  imaeq1i  4940  imaeq1d  4942  rneq  4957  f1imacnv  5303  clos1eq2  5876  eceq2  5964