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

Theorem oprssdmm 6167
Description: Domain of closure of an operation. (Contributed by Jim Kingdon, 23-Oct-2023.)
Hypotheses
Ref Expression
oprssdmm.m ((𝜑𝑢𝑆) → ∃𝑣 𝑣𝑢)
oprssdmm.cl ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
oprssdmm.f (𝜑 → Rel 𝐹)
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)

Proof of Theorem oprssdmm
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 elxp6 6165 . . . . . . 7 (𝑧 ∈ (𝑆 × 𝑆) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑆 ∧ (2nd𝑧) ∈ 𝑆)))
21biimpi 120 . . . . . 6 (𝑧 ∈ (𝑆 × 𝑆) → (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑆 ∧ (2nd𝑧) ∈ 𝑆)))
32adantl 277 . . . . 5 ((𝜑𝑧 ∈ (𝑆 × 𝑆)) → (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑆 ∧ (2nd𝑧) ∈ 𝑆)))
43simpld 112 . . . 4 ((𝜑𝑧 ∈ (𝑆 × 𝑆)) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
53simprd 114 . . . . 5 ((𝜑𝑧 ∈ (𝑆 × 𝑆)) → ((1st𝑧) ∈ 𝑆 ∧ (2nd𝑧) ∈ 𝑆))
6 oprssdmm.f . . . . . . . . 9 (𝜑 → Rel 𝐹)
76adantr 276 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → Rel 𝐹)
8 eleq2 2241 . . . . . . . . . 10 (𝑢 = (𝐹‘⟨𝑥, 𝑦⟩) → (𝑣𝑢𝑣 ∈ (𝐹‘⟨𝑥, 𝑦⟩)))
98exbidv 1825 . . . . . . . . 9 (𝑢 = (𝐹‘⟨𝑥, 𝑦⟩) → (∃𝑣 𝑣𝑢 ↔ ∃𝑣 𝑣 ∈ (𝐹‘⟨𝑥, 𝑦⟩)))
10 oprssdmm.m . . . . . . . . . . 11 ((𝜑𝑢𝑆) → ∃𝑣 𝑣𝑢)
1110ralrimiva 2550 . . . . . . . . . 10 (𝜑 → ∀𝑢𝑆𝑣 𝑣𝑢)
1211adantr 276 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → ∀𝑢𝑆𝑣 𝑣𝑢)
13 df-ov 5873 . . . . . . . . . 10 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
14 oprssdmm.cl . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
1513, 14eqeltrrid 2265 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝑆)
169, 12, 15rspcdva 2846 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → ∃𝑣 𝑣 ∈ (𝐹‘⟨𝑥, 𝑦⟩))
17 relelfvdm 5544 . . . . . . . . . 10 ((Rel 𝐹𝑣 ∈ (𝐹‘⟨𝑥, 𝑦⟩)) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
1817ex 115 . . . . . . . . 9 (Rel 𝐹 → (𝑣 ∈ (𝐹‘⟨𝑥, 𝑦⟩) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹))
1918exlimdv 1819 . . . . . . . 8 (Rel 𝐹 → (∃𝑣 𝑣 ∈ (𝐹‘⟨𝑥, 𝑦⟩) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹))
207, 16, 19sylc 62 . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
2120ralrimivva 2559 . . . . . 6 (𝜑 → ∀𝑥𝑆𝑦𝑆𝑥, 𝑦⟩ ∈ dom 𝐹)
2221adantr 276 . . . . 5 ((𝜑𝑧 ∈ (𝑆 × 𝑆)) → ∀𝑥𝑆𝑦𝑆𝑥, 𝑦⟩ ∈ dom 𝐹)
23 opeq1 3777 . . . . . . 7 (𝑥 = (1st𝑧) → ⟨𝑥, 𝑦⟩ = ⟨(1st𝑧), 𝑦⟩)
2423eleq1d 2246 . . . . . 6 (𝑥 = (1st𝑧) → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ ⟨(1st𝑧), 𝑦⟩ ∈ dom 𝐹))
25 opeq2 3778 . . . . . . 7 (𝑦 = (2nd𝑧) → ⟨(1st𝑧), 𝑦⟩ = ⟨(1st𝑧), (2nd𝑧)⟩)
2625eleq1d 2246 . . . . . 6 (𝑦 = (2nd𝑧) → (⟨(1st𝑧), 𝑦⟩ ∈ dom 𝐹 ↔ ⟨(1st𝑧), (2nd𝑧)⟩ ∈ dom 𝐹))
2724, 26rspc2va 2855 . . . . 5 ((((1st𝑧) ∈ 𝑆 ∧ (2nd𝑧) ∈ 𝑆) ∧ ∀𝑥𝑆𝑦𝑆𝑥, 𝑦⟩ ∈ dom 𝐹) → ⟨(1st𝑧), (2nd𝑧)⟩ ∈ dom 𝐹)
285, 22, 27syl2anc 411 . . . 4 ((𝜑𝑧 ∈ (𝑆 × 𝑆)) → ⟨(1st𝑧), (2nd𝑧)⟩ ∈ dom 𝐹)
294, 28eqeltrd 2254 . . 3 ((𝜑𝑧 ∈ (𝑆 × 𝑆)) → 𝑧 ∈ dom 𝐹)
3029ex 115 . 2 (𝜑 → (𝑧 ∈ (𝑆 × 𝑆) → 𝑧 ∈ dom 𝐹))
3130ssrdv 3161 1 (𝜑 → (𝑆 × 𝑆) ⊆ dom 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wex 1492  wcel 2148  wral 2455  wss 3129  cop 3595   × cxp 4622  dom cdm 4624  Rel wrel 4629  cfv 5213  (class class class)co 5870  1st c1st 6134  2nd c2nd 6135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4119  ax-pow 4172  ax-pr 4207  ax-un 4431
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3809  df-br 4002  df-opab 4063  df-mpt 4064  df-id 4291  df-xp 4630  df-rel 4631  df-cnv 4632  df-co 4633  df-dm 4634  df-rn 4635  df-iota 5175  df-fun 5215  df-fv 5221  df-ov 5873  df-1st 6136  df-2nd 6137
This theorem is referenced by:  axaddf  7862  axmulf  7863
  Copyright terms: Public domain W3C validator