Theorem ndmaovcl 48242

Description: The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 7604 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 5687	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
3		ndmaov.1	. . . . . 6 ⊢ dom 𝐹 = (𝑆 × 𝑆)
4	3	eqcomi 2770	. . . . 5 ⊢ (𝑆 × 𝑆) = dom 𝐹
5	4	eleq2i 2853	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
6		ndmaovcl.3	. . . . 5 ⊢ ((𝐴𝐹𝐵)) ∈ V
7		ndmaov 48222	. . . . 5 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V)
8		eleq1 2849	. . . . . . 7 ⊢ ( ((𝐴𝐹𝐵)) = V → ( ((𝐴𝐹𝐵)) ∈ V ↔ V ∈ V))
9	8	biimpd 232	. . . . . 6 ⊢ ( ((𝐴𝐹𝐵)) = V → ( ((𝐴𝐹𝐵)) ∈ V → V ∈ V))
10		vprc 5274	. . . . . . 7 ⊢ ¬ V ∈ V
11	10	pm2.21i 120	. . . . . 6 ⊢ (V ∈ V → ((𝐴𝐹𝐵)) ∈ 𝑆)
12	9, 11	syl6com 38	. . . . 5 ⊢ ( ((𝐴𝐹𝐵)) ∈ V → ( ((𝐴𝐹𝐵)) = V → ((𝐴𝐹𝐵)) ∈ 𝑆))
13	6, 7, 12	mpsyl 69	. . . 4 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) ∈ 𝑆)
14	5, 13	sylnbi 333	. . 3 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆)
15	2, 14	sylnbir 334	. 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆)
16	1, 15	pm2.61i 184	1 ⊢ ((𝐴𝐹𝐵)) ∈ 𝑆

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ⟨cop 4590   × cxp 5649  dom cdm 5651   ((caov 48157

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-res 5663  df-iota 6493  df-fun 6539  df-fv 6545  df-aiota 48124  df-dfat 48158  df-afv 48159  df-aov 48160

This theorem is used by: (None)