Theorem imaeq1 5380

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

Assertion

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

Proof of Theorem imaeq1

Step	Hyp	Ref	Expression
1		reseq1 5311	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶))
2	1	rneqd 5274	. 2 ⊢ (𝐴 = 𝐵 → ran (𝐴 ↾ 𝐶) = ran (𝐵 ↾ 𝐶))
3		df-ima 5051	. 2 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶)
4		df-ima 5051	. 2 ⊢ (𝐵 “ 𝐶) = ran (𝐵 ↾ 𝐶)
5	2, 3, 4	3eqtr4g 2669	1 ⊢ (𝐴 = 𝐵 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶))

Syntax hints:   → wi 4   = wceq 1475  ran crn 5039   ↾ cres 5040   “ cima 5041

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051

This theorem is referenced by:  imaeq1i  5382  imaeq1d  5384  suppval  7184  eceq2  7671  marypha1lem  8222  marypha1  8223  ackbij2lem2  8945  ackbij2lem3  8946  r1om  8949  limsupval  14053  isacs1i  16141  mreacs  16142  islindf  19970  iscnp  20851  xkoccn  21232  xkohaus  21266  xkoco1cn  21270  xkoco2cn  21271  xkococnlem  21272  xkococn  21273  xkoinjcn  21300  fmval  21557  fmf  21559  utoptop  21848  restutop  21851  restutopopn  21852  ustuqtoplem  21853  ustuqtop1  21855  ustuqtop2  21856  ustuqtop4  21858  ustuqtop5  21859  utopsnneiplem  21861  utopsnnei  21863  neipcfilu  21910  psmetutop  22182  cfilfval  22870  elply2  23756  coeeu  23785  coelem  23786  coeeq  23787  dmarea  24484  mclsax  30720  tailfval  31537  bj-cleq  32142  poimirlem15  32594  poimirlem24  32603  brtrclfv2  37038