Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ndmaovcl Structured version   Visualization version   GIF version

Theorem ndmaovcl 42838
Description: The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 7148 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 ((𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆)
ndmaovcl.3 ((𝐴𝐹𝐵)) ∈ V
Assertion
Ref Expression
ndmaovcl ((𝐴𝐹𝐵)) ∈ 𝑆

Proof of Theorem ndmaovcl
StepHypRef Expression
1 ndmaovcl.2 . 2 ((𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆)
2 opelxp 5440 . . 3 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆𝐵𝑆))
3 ndmaov.1 . . . . . 6 dom 𝐹 = (𝑆 × 𝑆)
43eqcomi 2782 . . . . 5 (𝑆 × 𝑆) = dom 𝐹
54eleq2i 2852 . . . 4 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
6 ndmaovcl.3 . . . . 5 ((𝐴𝐹𝐵)) ∈ V
7 ndmaov 42818 . . . . 5 (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V)
8 eleq1 2848 . . . . . . 7 ( ((𝐴𝐹𝐵)) = V → ( ((𝐴𝐹𝐵)) ∈ V ↔ V ∈ V))
98biimpd 221 . . . . . 6 ( ((𝐴𝐹𝐵)) = V → ( ((𝐴𝐹𝐵)) ∈ V → V ∈ V))
10 vprc 5073 . . . . . . 7 ¬ V ∈ V
1110pm2.21i 117 . . . . . 6 (V ∈ V → ((𝐴𝐹𝐵)) ∈ 𝑆)
129, 11syl6com 37 . . . . 5 ( ((𝐴𝐹𝐵)) ∈ V → ( ((𝐴𝐹𝐵)) = V → ((𝐴𝐹𝐵)) ∈ 𝑆))
136, 7, 12mpsyl 68 . . . 4 (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) ∈ 𝑆)
145, 13sylnbi 322 . . 3 (¬ ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆)
152, 14sylnbir 323 . 2 (¬ (𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵)) ∈ 𝑆)
161, 15pm2.61i 177 1 ((𝐴𝐹𝐵)) ∈ 𝑆
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 387   = wceq 1508  wcel 2051  Vcvv 3410  cop 4442   × cxp 5402  dom cdm 5404   ((caov 42753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-fal 1521  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-ral 3088  df-rex 3089  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-int 4747  df-br 4927  df-opab 4989  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-res 5416  df-iota 6150  df-fun 6188  df-fv 6194  df-aiota 42721  df-dfat 42754  df-afv 42755  df-aov 42756
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator