Theorem rneqd 2557

Description: Equality deduction for range.

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 2555	. 2 ⊢ (A = B → ran A = ran B)
3	1, 2	syl 12	1 ⊢ (φ → ran A = ran B)

Syntax hints:   → wi 2   = wceq 1091  ran crn 2411

This theorem is referenced by:  imaeq1 2602  imaeq2 2603  elxp4 2640  elxp5 2641  funimacnv 2711  2ndval 3090  fo2nd 3095  en1 3331  xpassen 3344  xpdom2 3345  sbthlem4 3352  xpmapenlem2 3392  xpmapenlem4 3394  xpmapenlem5 3395  mapunen 3397  fodomb 3615  xpnnen 4927  pj3 5660

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-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-cnv 2426  df-dm 2428  df-rn 2429

metamath.org