Theorem aovfundmoveq 47946

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 47903	. 2 ⊢ (𝐹 defAt ⟨𝐴, 𝐵⟩ → (𝐹'''⟨𝐴, 𝐵⟩) = (𝐹‘⟨𝐴, 𝐵⟩))
2		df-aov 47886	. 2 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
3		df-ov 7413	. 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
4	1, 2, 3	3eqtr4g 2823	1 ⊢ (𝐹 defAt ⟨𝐴, 𝐵⟩ → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ⟨cop 4595  ‘cfv 6536  (class class class)co 7410   defAt wdfat 47881  '''cafv 47882   ((caov 47883

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-res 5673  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-aiota 47850  df-dfat 47884  df-afv 47885  df-aov 47886

This theorem is used by:  aovmpt4g  47966