Theorem aovovn0oveq 47318

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 7355	. . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2	1	neeq1i 2993	. 2 ⊢ ((𝐴𝐹𝐵) ≠ ∅ ↔ (𝐹‘⟨𝐴, 𝐵⟩) ≠ ∅)
3		afvfvn0fveq 47274	. . 3 ⊢ ((𝐹‘⟨𝐴, 𝐵⟩) ≠ ∅ → (𝐹'''⟨𝐴, 𝐵⟩) = (𝐹‘⟨𝐴, 𝐵⟩))
4		df-aov 47245	. . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
5	3, 4, 1	3eqtr4g 2793	. 2 ⊢ ((𝐹‘⟨𝐴, 𝐵⟩) ≠ ∅ → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))
6	2, 5	sylbi 217	1 ⊢ ((𝐴𝐹𝐵) ≠ ∅ → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))

Syntax hints:   → wi 4   = wceq 1541   ≠ wne 2929  ∅c0 4282  ⟨cop 4581  ‘cfv 6486  (class class class)co 7352  '''cafv 47241   ((caov 47242

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-res 5631  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7355  df-aiota 47209  df-dfat 47243  df-afv 47244  df-aov 47245

This theorem is referenced by: (None)