Theorem ndmaovcl 47666

Description: The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 7541 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 5654	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
3		ndmaov.1	. . . . . 6 ⊢ dom 𝐹 = (𝑆 × 𝑆)
4	3	eqcomi 2748	. . . . 5 ⊢ (𝑆 × 𝑆) = dom 𝐹
5	4	eleq2i 2831	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
6		ndmaovcl.3	. . . . 5 ⊢ ((𝐴𝐹𝐵)) ∈ V
7		ndmaov 47646	. . . . 5 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V)
8		eleq1 2827	. . . . . . 7 ⊢ ( ((𝐴𝐹𝐵)) = V → ( ((𝐴𝐹𝐵)) ∈ V ↔ V ∈ V))
9	8	biimpd 230	. . . . . 6 ⊢ ( ((𝐴𝐹𝐵)) = V → ( ((𝐴𝐹𝐵)) ∈ V → V ∈ V))
10		vprc 5242	. . . . . . 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 331	. . 3 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆)
15	2, 14	sylnbir 332	. 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆)
16	1, 15	pm2.61i 183	1 ⊢ ((𝐴𝐹𝐵)) ∈ 𝑆

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3431  ⟨cop 4561   × cxp 5616  dom cdm 5618   ((caov 47581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-res 5630  df-iota 6441  df-fun 6487  df-fv 6493  df-aiota 47548  df-dfat 47582  df-afv 47583  df-aov 47584

This theorem is referenced by: (None)