Theorem ndmaovcl 47208

Description: The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 7577 where it is required that the domain contains the empty set (∅ ∈ 𝑆). (Contributed by Alexander van der Vekens, 26-May-2017.)

Hypotheses

Ref	Expression
ndmaov.1	⊢ dom 𝐹 = (𝑆 × 𝑆)
ndmaovcl.2	⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆)
ndmaovcl.3	⊢ ((𝐴𝐹𝐵)) ∈ V

Assertion

Ref	Expression
ndmaovcl	⊢ ((𝐴𝐹𝐵)) ∈ 𝑆

Proof of Theorem ndmaovcl

Step	Hyp	Ref	Expression
1		ndmaovcl.2	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆)
2		opelxp 5677	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
3		ndmaov.1	. . . . . 6 ⊢ dom 𝐹 = (𝑆 × 𝑆)
4	3	eqcomi 2739	. . . . 5 ⊢ (𝑆 × 𝑆) = dom 𝐹
5	4	eleq2i 2821	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
6		ndmaovcl.3	. . . . 5 ⊢ ((𝐴𝐹𝐵)) ∈ V
7		ndmaov 47188	. . . . 5 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V)
8		eleq1 2817	. . . . . . 7 ⊢ ( ((𝐴𝐹𝐵)) = V → ( ((𝐴𝐹𝐵)) ∈ V ↔ V ∈ V))
9	8	biimpd 229	. . . . . 6 ⊢ ( ((𝐴𝐹𝐵)) = V → ( ((𝐴𝐹𝐵)) ∈ V → V ∈ V))
10		vprc 5273	. . . . . . 7 ⊢ ¬ V ∈ V
11	10	pm2.21i 119	. . . . . 6 ⊢ (V ∈ V → ((𝐴𝐹𝐵)) ∈ 𝑆)
12	9, 11	syl6com 37	. . . . 5 ⊢ ( ((𝐴𝐹𝐵)) ∈ V → ( ((𝐴𝐹𝐵)) = V → ((𝐴𝐹𝐵)) ∈ 𝑆))
13	6, 7, 12	mpsyl 68	. . . 4 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) ∈ 𝑆)
14	5, 13	sylnbi 330	. . 3 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆)
15	2, 14	sylnbir 331	. 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆)
16	1, 15	pm2.61i 182	1 ⊢ ((𝐴𝐹𝐵)) ∈ 𝑆

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450  ⟨cop 4598   × cxp 5639  dom cdm 5641   ((caov 47123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-res 5653  df-iota 6467  df-fun 6516  df-fv 6522  df-aiota 47090  df-dfat 47124  df-afv 47125  df-aov 47126

This theorem is referenced by: (None)