Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  oprssdmm GIF version

 Description: Domain of closure of an operation. (Contributed by Jim Kingdon, 23-Oct-2023.)
Hypotheses
Ref Expression
oprssdmm.cl ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
Assertion
Ref Expression
oprssdmm (𝜑 → (𝑆 × 𝑆) ⊆ dom 𝐹)
Distinct variable groups:   𝑢,𝐹,𝑣,𝑥,𝑦   𝑢,𝑆,𝑥,𝑦   𝜑,𝑢,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑣)   𝑆(𝑣)

Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 elxp6 6111 . . . . . . 7 (𝑧 ∈ (𝑆 × 𝑆) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑆 ∧ (2nd𝑧) ∈ 𝑆)))
21biimpi 119 . . . . . 6 (𝑧 ∈ (𝑆 × 𝑆) → (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑆 ∧ (2nd𝑧) ∈ 𝑆)))
32adantl 275 . . . . 5 ((𝜑𝑧 ∈ (𝑆 × 𝑆)) → (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑆 ∧ (2nd𝑧) ∈ 𝑆)))
43simpld 111 . . . 4 ((𝜑𝑧 ∈ (𝑆 × 𝑆)) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
53simprd 113 . . . . 5 ((𝜑𝑧 ∈ (𝑆 × 𝑆)) → ((1st𝑧) ∈ 𝑆 ∧ (2nd𝑧) ∈ 𝑆))
6 oprssdmm.f . . . . . . . . 9 (𝜑 → Rel 𝐹)
76adantr 274 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → Rel 𝐹)
8 eleq2 2221 . . . . . . . . . 10 (𝑢 = (𝐹‘⟨𝑥, 𝑦⟩) → (𝑣𝑢𝑣 ∈ (𝐹‘⟨𝑥, 𝑦⟩)))
98exbidv 1805 . . . . . . . . 9 (𝑢 = (𝐹‘⟨𝑥, 𝑦⟩) → (∃𝑣 𝑣𝑢 ↔ ∃𝑣 𝑣 ∈ (𝐹‘⟨𝑥, 𝑦⟩)))
10 oprssdmm.m . . . . . . . . . . 11 ((𝜑𝑢𝑆) → ∃𝑣 𝑣𝑢)
1110ralrimiva 2530 . . . . . . . . . 10 (𝜑 → ∀𝑢𝑆𝑣 𝑣𝑢)
1211adantr 274 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → ∀𝑢𝑆𝑣 𝑣𝑢)
13 df-ov 5821 . . . . . . . . . 10 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
14 oprssdmm.cl . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
1513, 14eqeltrrid 2245 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝑆)
169, 12, 15rspcdva 2821 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → ∃𝑣 𝑣 ∈ (𝐹‘⟨𝑥, 𝑦⟩))
17 relelfvdm 5497 . . . . . . . . . 10 ((Rel 𝐹𝑣 ∈ (𝐹‘⟨𝑥, 𝑦⟩)) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
1817ex 114 . . . . . . . . 9 (Rel 𝐹 → (𝑣 ∈ (𝐹‘⟨𝑥, 𝑦⟩) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹))
1918exlimdv 1799 . . . . . . . 8 (Rel 𝐹 → (∃𝑣 𝑣 ∈ (𝐹‘⟨𝑥, 𝑦⟩) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹))
207, 16, 19sylc 62 . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
2120ralrimivva 2539 . . . . . 6 (𝜑 → ∀𝑥𝑆𝑦𝑆𝑥, 𝑦⟩ ∈ dom 𝐹)
2221adantr 274 . . . . 5 ((𝜑𝑧 ∈ (𝑆 × 𝑆)) → ∀𝑥𝑆𝑦𝑆𝑥, 𝑦⟩ ∈ dom 𝐹)
23 opeq1 3741 . . . . . . 7 (𝑥 = (1st𝑧) → ⟨𝑥, 𝑦⟩ = ⟨(1st𝑧), 𝑦⟩)
2423eleq1d 2226 . . . . . 6 (𝑥 = (1st𝑧) → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ ⟨(1st𝑧), 𝑦⟩ ∈ dom 𝐹))
25 opeq2 3742 . . . . . . 7 (𝑦 = (2nd𝑧) → ⟨(1st𝑧), 𝑦⟩ = ⟨(1st𝑧), (2nd𝑧)⟩)
2625eleq1d 2226 . . . . . 6 (𝑦 = (2nd𝑧) → (⟨(1st𝑧), 𝑦⟩ ∈ dom 𝐹 ↔ ⟨(1st𝑧), (2nd𝑧)⟩ ∈ dom 𝐹))
2724, 26rspc2va 2830 . . . . 5 ((((1st𝑧) ∈ 𝑆 ∧ (2nd𝑧) ∈ 𝑆) ∧ ∀𝑥𝑆𝑦𝑆𝑥, 𝑦⟩ ∈ dom 𝐹) → ⟨(1st𝑧), (2nd𝑧)⟩ ∈ dom 𝐹)
285, 22, 27syl2anc 409 . . . 4 ((𝜑𝑧 ∈ (𝑆 × 𝑆)) → ⟨(1st𝑧), (2nd𝑧)⟩ ∈ dom 𝐹)
294, 28eqeltrd 2234 . . 3 ((𝜑𝑧 ∈ (𝑆 × 𝑆)) → 𝑧 ∈ dom 𝐹)
3029ex 114 . 2 (𝜑 → (𝑧 ∈ (𝑆 × 𝑆) → 𝑧 ∈ dom 𝐹))
3130ssrdv 3134 1 (𝜑 → (𝑆 × 𝑆) ⊆ dom 𝐹)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1335  ∃wex 1472   ∈ wcel 2128  ∀wral 2435   ⊆ wss 3102  ⟨cop 3563   × cxp 4581  dom cdm 4583  Rel wrel 4588  ‘cfv 5167  (class class class)co 5818  1st c1st 6080  2nd c2nd 6081 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168  ax-un 4392 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4252  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-iota 5132  df-fun 5169  df-fv 5175  df-ov 5821  df-1st 6082  df-2nd 6083 This theorem is referenced by:  axaddf  7771  axmulf  7772
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