Theorem ndmaovrcl 47314

Description: Reverse closure law, in contrast to ndmovrcl 7532 where it is required that the operation's domain doesn't contain the empty set (¬ ∅ ∈ 𝑆), no additional asumption is required. (Contributed by Alexander van der Vekens, 26-May-2017.)

Hypothesis

Ref	Expression
ndmaov.1	⊢ dom 𝐹 = (𝑆 × 𝑆)

Assertion

Ref	Expression
ndmaovrcl	⊢ ( ((𝐴𝐹𝐵)) ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))

Proof of Theorem ndmaovrcl

Step	Hyp	Ref	Expression
1		aovvdm 47295	. 2 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝑆 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
2		opelxp 5650	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
3	2	biimpi 216	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
4		ndmaov.1	. . 3 ⊢ dom 𝐹 = (𝑆 × 𝑆)
5	3, 4	eleq2s 2849	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
6	1, 5	syl 17	1 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ⟨cop 4579   × cxp 5612  dom cdm 5614   ((caov 47228

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-res 5626  df-iota 6437  df-fun 6483  df-fv 6489  df-aiota 47195  df-dfat 47229  df-afv 47230  df-aov 47231

This theorem is referenced by: (None)