Theorem imaeq1 6006

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 5924	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶))
2	1	rneqd 5880	. 2 ⊢ (𝐴 = 𝐵 → ran (𝐴 ↾ 𝐶) = ran (𝐵 ↾ 𝐶))
3		df-ima 5632	. 2 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶)
4		df-ima 5632	. 2 ⊢ (𝐵 “ 𝐶) = ran (𝐵 ↾ 𝐶)
5	2, 3, 4	3eqtr4g 2789	1 ⊢ (𝐴 = 𝐵 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶))

Syntax hints:   → wi 4   = wceq 1540  ran crn 5620   ↾ cres 5621   “ cima 5622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-cnv 5627  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632

This theorem is referenced by:  imaeq1i  6008  imaeq1d  6010  suppval  8095  naddcllem  8594  eceq2  8666  marypha1lem  9323  marypha1  9324  ackbij2lem2  10133  ackbij2lem3  10134  r1om  10137  limsupval  15381  isacs1i  17563  mreacs  17564  islindf  21719  iscnp  23122  xkoccn  23504  xkohaus  23538  xkoco1cn  23542  xkoco2cn  23543  xkococnlem  23544  xkococn  23545  xkoinjcn  23572  fmval  23828  fmf  23830  utoptop  24120  restutop  24123  restutopopn  24124  ustuqtoplem  24125  ustuqtop1  24127  ustuqtop2  24128  ustuqtop4  24130  ustuqtop5  24131  utopsnneiplem  24133  utopsnnei  24135  neipcfilu  24181  psmetutop  24453  cfilfval  25162  elply2  26099  coeeu  26128  coelem  26129  coeeq  26130  dmarea  26865  negsval  27936  mclsax  35542  tailfval  36346  bj-cleq  36936  bj-funun  37226  poimirlem15  37615  poimirlem24  37624  brtrclfv2  43700  liminfval  45740  ushggricedg  47911  uhgrimisgrgric  47915