Theorem aovovn0oveq 46360

Description: If the operation's value at an argument is not the empty set, it equals the value of the alternative operation at this argument. (Contributed by Alexander van der Vekens, 26-May-2017.)

Assertion

Ref	Expression
aovovn0oveq	⊢ ((𝐴𝐹𝐵) ≠ ∅ → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))

Proof of Theorem aovovn0oveq

Step	Hyp	Ref	Expression
1		df-ov 7415	. . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2	1	neeq1i 3004	. 2 ⊢ ((𝐴𝐹𝐵) ≠ ∅ ↔ (𝐹‘⟨𝐴, 𝐵⟩) ≠ ∅)
3		afvfvn0fveq 46316	. . 3 ⊢ ((𝐹‘⟨𝐴, 𝐵⟩) ≠ ∅ → (𝐹'''⟨𝐴, 𝐵⟩) = (𝐹‘⟨𝐴, 𝐵⟩))
4		df-aov 46287	. . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
5	3, 4, 1	3eqtr4g 2796	. 2 ⊢ ((𝐹‘⟨𝐴, 𝐵⟩) ≠ ∅ → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))
6	2, 5	sylbi 216	1 ⊢ ((𝐴𝐹𝐵) ≠ ∅ → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))

Syntax hints:   → wi 4   = wceq 1540   ≠ wne 2939  ∅c0 4322  ⟨cop 4634  ‘cfv 6543  (class class class)co 7412  '''cafv 46283   ((caov 46284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-res 5688  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-aiota 46251  df-dfat 46285  df-afv 46286  df-aov 46287

This theorem is referenced by: (None)