Theorem ndmaovcl 47828

Description: The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 7596 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 5698	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
3		ndmaov.1	. . . . . 6 ⊢ dom 𝐹 = (𝑆 × 𝑆)
4	3	eqcomi 2778	. . . . 5 ⊢ (𝑆 × 𝑆) = dom 𝐹
5	4	eleq2i 2861	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
6		ndmaovcl.3	. . . . 5 ⊢ ((𝐴𝐹𝐵)) ∈ V
7		ndmaov 47808	. . . . 5 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V)
8		eleq1 2857	. . . . . . 7 ⊢ ( ((𝐴𝐹𝐵)) = V → ( ((𝐴𝐹𝐵)) ∈ V ↔ V ∈ V))
9	8	biimpd 232	. . . . . 6 ⊢ ( ((𝐴𝐹𝐵)) = V → ( ((𝐴𝐹𝐵)) ∈ V → V ∈ V))
10		vprc 5285	. . . . . . 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

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ⟨cop 4600   × cxp 5660  dom cdm 5662   ((caov 47743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-res 5674  df-iota 6493  df-fun 6539  df-fv 6545  df-aiota 47710  df-dfat 47744  df-afv 47745  df-aov 47746

This theorem is referenced by: (None)