Theorem imaeq1 6020

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 5938	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶))
2	1	rneqd 5893	. 2 ⊢ (𝐴 = 𝐵 → ran (𝐴 ↾ 𝐶) = ran (𝐵 ↾ 𝐶))
3		df-ima 5644	. 2 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶)
4		df-ima 5644	. 2 ⊢ (𝐵 “ 𝐶) = ran (𝐵 ↾ 𝐶)
5	2, 3, 4	3eqtr4g 2796	1 ⊢ (𝐴 = 𝐵 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶))

Syntax hints:   → wi 4   = wceq 1542  ran crn 5632   ↾ cres 5633   “ cima 5634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644

This theorem is referenced by:  imaeq1i  6022  imaeq1d  6024  suppval  8112  naddcllem  8612  eceq2  8685  marypha1lem  9346  marypha1  9347  ackbij2lem2  10161  ackbij2lem3  10162  r1om  10165  limsupval  15436  isacs1i  17623  mreacs  17624  islindf  21792  iscnp  23202  xkoccn  23584  xkohaus  23618  xkoco1cn  23622  xkoco2cn  23623  xkococnlem  23624  xkococn  23625  xkoinjcn  23652  fmval  23908  fmf  23910  utoptop  24199  restutop  24202  restutopopn  24203  ustuqtoplem  24204  ustuqtop1  24206  ustuqtop2  24207  ustuqtop4  24209  ustuqtop5  24210  utopsnneiplem  24212  utopsnnei  24214  neipcfilu  24260  psmetutop  24532  cfilfval  25231  elply2  26161  coeeu  26190  coelem  26191  coeeq  26192  dmarea  26921  negsval  28017  mclsax  35751  tailfval  36554  bj-cleq  37269  bj-funun  37566  poimirlem15  37956  poimirlem24  37965  brtrclfv2  44154  liminfval  46187  ushggricedg  48403  uhgrimisgrgric  48407