Theorem rneq 4911

Description: Equality theorem for range. (Contributed by NM, 29-Dec-1996.)

Assertion

Ref	Expression
rneq	⊢ (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)

Proof of Theorem rneq

Step	Hyp	Ref	Expression
1		cnveq 4857	. . 3 ⊢ (𝐴 = 𝐵 → ◡𝐴 = ◡𝐵)
2	1	dmeqd 4886	. 2 ⊢ (𝐴 = 𝐵 → dom ◡𝐴 = dom ◡𝐵)
3		df-rn 4691	. 2 ⊢ ran 𝐴 = dom ◡𝐴
4		df-rn 4691	. 2 ⊢ ran 𝐵 = dom ◡𝐵
5	2, 3, 4	3eqtr4g 2264	1 ⊢ (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)

Syntax hints:   → wi 4   = wceq 1373  ^◡ccnv 4679  dom cdm 4680  ran crn 4681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3172  df-in 3174  df-ss 3181  df-sn 3641  df-pr 3642  df-op 3644  df-br 4049  df-opab 4111  df-cnv 4688  df-dm 4690  df-rn 4691

This theorem is referenced by:  rneqi  4912  rneqd  4913  xpima1  5135  feq1  5415  foeq1  5503  ixpsnf1o  6833  imasex  13187