Theorem aovfundmoveq 47644

Description: If a class is a function restricted to an ordered pair of its domain, then the value of the operation on this pair is equal for both definitions. (Contributed by Alexander van der Vekens, 26-May-2017.)

Assertion

Ref	Expression
aovfundmoveq	⊢ (𝐹 defAt ⟨𝐴, 𝐵⟩ → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))

Proof of Theorem aovfundmoveq

Step	Hyp	Ref	Expression
1		afvfundmfveq 47601	. 2 ⊢ (𝐹 defAt ⟨𝐴, 𝐵⟩ → (𝐹'''⟨𝐴, 𝐵⟩) = (𝐹‘⟨𝐴, 𝐵⟩))
2		df-aov 47584	. 2 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
3		df-ov 7364	. 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
4	1, 2, 3	3eqtr4g 2797	1 ⊢ (𝐹 defAt ⟨𝐴, 𝐵⟩ → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))

Syntax hints:   → wi 4   = wceq 1542  ⟨cop 4574  ‘cfv 6493  (class class class)co 7361   defAt wdfat 47579  '''cafv 47580   ((caov 47581

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-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371

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-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-br 5087  df-opab 5149  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-res 5637  df-iota 6449  df-fun 6495  df-fv 6501  df-ov 7364  df-aiota 47548  df-dfat 47582  df-afv 47583  df-aov 47584

This theorem is referenced by:  aovmpt4g  47664