Theorem rneqd 4959

Description: Equality deduction for range. (Contributed by set.mm contributors, 4-Mar-2004.)

Hypothesis

Ref	Expression
rneqd.1	⊢ (φ → A = B)

Assertion

Ref	Expression
rneqd	⊢ (φ → ran A = ran B)

Proof of Theorem rneqd

Step	Hyp	Ref	Expression
1		rneqd.1	. 2 ⊢ (φ → A = B)
2		rneq 4957	. 2 ⊢ (A = B → ran A = ran B)
3	1, 2	syl 15	1 ⊢ (φ → ran A = ran B)

Syntax hints:   → wi 4   = wceq 1642  ran crn 4774

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  df-rn 4787

This theorem is referenced by:  resima2  5008  resiima  5013  rnxpid  5055  elxp4  5109  funimacnv  5169  fnima  5202