Theorem foeq3 6790

Description: Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)

Assertion

Ref	Expression
foeq3	⊢ (𝐴 = 𝐵 → (𝐹:𝐶–onto→𝐴 ↔ 𝐹:𝐶–onto→𝐵))

Proof of Theorem foeq3

Step	Hyp	Ref	Expression
1		eqeq2 2774	. . 3 ⊢ (𝐴 = 𝐵 → (ran 𝐹 = 𝐴 ↔ ran 𝐹 = 𝐵))
2	1	anbi2d 641	. 2 ⊢ (𝐴 = 𝐵 → ((𝐹 Fn 𝐶 ∧ ran 𝐹 = 𝐴) ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹 = 𝐵)))
3		df-fo 6542	. 2 ⊢ (𝐹:𝐶–onto→𝐴 ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹 = 𝐴))
4		df-fo 6542	. 2 ⊢ (𝐹:𝐶–onto→𝐵 ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹 = 𝐵))
5	2, 3, 4	3bitr4g 317	1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶–onto→𝐴 ↔ 𝐹:𝐶–onto→𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569  ran crn 5661   Fn wfn 6531  –onto→wfo 6534

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-cleq 2754  df-fo 6542

This theorem is used by:  fimadmfo  6801  f1oeq3  6810  foeq123d  6813  resdif  6842  ncanth  7367  ffoss  7941  rneqdmfinf1o  9288  fidomdm  9289  fifo  9390  brwdom  9527  brwdom2  9533  canthwdom  9539  ixpiunwdom  9550  fin1a2lem7  10396  dmct  10514  s7f1o  15010  znnen  16274  quslem  17603  znzrhfo  21708  rncmp  23564  connima  23593  conncn  23594  qtopcmplem  23875  qtoprest  23885  eupths  30562  pjhfo  32069  2ndresdjuf1o  33006  cycpmconjvlem  33470  algextdeglem8  34123  msrfo  36046  ivthALT  36874  bj-inftyexpitaufo  37874  poimirlem26  38325  poimirlem27  38326  opidon2OLD  38533  founiiun0  45936  focofob  47845  fundcmpsurinj  48186  fundcmpsurbijinj  48187  imasubc  49957  fullthinc  50256