Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  imasaddfnlem Structured version   Visualization version   GIF version

 Description: The image structure operation is a function if the original operation is compatible with the function. (Contributed by Mario Carneiro, 23-Feb-2015.)
Hypotheses
Ref Expression
imasaddf.e ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))
imasaddflem.a (𝜑 = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
Assertion
Ref Expression
imasaddfnlem (𝜑 Fn (𝐵 × 𝐵))
Distinct variable groups:   𝑞,𝑝,𝐵   𝑎,𝑏,𝑝,𝑞,𝑉   · ,𝑝,𝑞   𝐹,𝑎,𝑏,𝑝,𝑞   𝜑,𝑎,𝑏,𝑝,𝑞   ,𝑎,𝑏,𝑝,𝑞
Allowed substitution hints:   𝐵(𝑎,𝑏)   · (𝑎,𝑏)

Dummy variables 𝑤 𝑦 𝑧 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opex 5321 . . . . . . . . 9 ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ V
2 fvex 6658 . . . . . . . . 9 (𝐹‘(𝑝 · 𝑞)) ∈ V
31, 2relsnop 5642 . . . . . . . 8 Rel {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩}
43rgenw 3118 . . . . . . 7 𝑞𝑉 Rel {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩}
5 reliun 5653 . . . . . . 7 (Rel 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ↔ ∀𝑞𝑉 Rel {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
64, 5mpbir 234 . . . . . 6 Rel 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩}
76rgenw 3118 . . . . 5 𝑝𝑉 Rel 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩}
8 reliun 5653 . . . . 5 (Rel 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ↔ ∀𝑝𝑉 Rel 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
97, 8mpbir 234 . . . 4 Rel 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩}
10 imasaddflem.a . . . . 5 (𝜑 = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
1110releqd 5617 . . . 4 (𝜑 → (Rel ↔ Rel 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩}))
129, 11mpbiri 261 . . 3 (𝜑 → Rel )
13 imasaddf.f . . . . . . . . . . . . . . . 16 (𝜑𝐹:𝑉onto𝐵)
14 fof 6565 . . . . . . . . . . . . . . . 16 (𝐹:𝑉onto𝐵𝐹:𝑉𝐵)
1513, 14syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐹:𝑉𝐵)
16 ffvelrn 6826 . . . . . . . . . . . . . . . 16 ((𝐹:𝑉𝐵𝑝𝑉) → (𝐹𝑝) ∈ 𝐵)
17 ffvelrn 6826 . . . . . . . . . . . . . . . 16 ((𝐹:𝑉𝐵𝑞𝑉) → (𝐹𝑞) ∈ 𝐵)
1816, 17anim12dan 621 . . . . . . . . . . . . . . 15 ((𝐹:𝑉𝐵 ∧ (𝑝𝑉𝑞𝑉)) → ((𝐹𝑝) ∈ 𝐵 ∧ (𝐹𝑞) ∈ 𝐵))
1915, 18sylan 583 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → ((𝐹𝑝) ∈ 𝐵 ∧ (𝐹𝑞) ∈ 𝐵))
20 opelxpi 5556 . . . . . . . . . . . . . 14 (((𝐹𝑝) ∈ 𝐵 ∧ (𝐹𝑞) ∈ 𝐵) → ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ (𝐵 × 𝐵))
2119, 20syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ (𝐵 × 𝐵))
22 opelxpi 5556 . . . . . . . . . . . . 13 ((⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ (𝐵 × 𝐵) ∧ (𝐹‘(𝑝 · 𝑞)) ∈ V) → ⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ ∈ ((𝐵 × 𝐵) × V))
2321, 2, 22sylancl 589 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → ⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ ∈ ((𝐵 × 𝐵) × V))
2423snssd 4702 . . . . . . . . . . 11 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × V))
2524anassrs 471 . . . . . . . . . 10 (((𝜑𝑝𝑉) ∧ 𝑞𝑉) → {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × V))
2625iunssd 4937 . . . . . . . . 9 ((𝜑𝑝𝑉) → 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × V))
2726iunssd 4937 . . . . . . . 8 (𝜑 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × V))
2810, 27eqsstrd 3953 . . . . . . 7 (𝜑 ⊆ ((𝐵 × 𝐵) × V))
29 dmss 5735 . . . . . . 7 ( ⊆ ((𝐵 × 𝐵) × V) → dom ⊆ dom ((𝐵 × 𝐵) × V))
3028, 29syl 17 . . . . . 6 (𝜑 → dom ⊆ dom ((𝐵 × 𝐵) × V))
31 vn0 4254 . . . . . . 7 V ≠ ∅
32 dmxp 5763 . . . . . . 7 (V ≠ ∅ → dom ((𝐵 × 𝐵) × V) = (𝐵 × 𝐵))
3331, 32ax-mp 5 . . . . . 6 dom ((𝐵 × 𝐵) × V) = (𝐵 × 𝐵)
3430, 33sseqtrdi 3965 . . . . 5 (𝜑 → dom ⊆ (𝐵 × 𝐵))
35 forn 6568 . . . . . . 7 (𝐹:𝑉onto𝐵 → ran 𝐹 = 𝐵)
3613, 35syl 17 . . . . . 6 (𝜑 → ran 𝐹 = 𝐵)
3736sqxpeqd 5551 . . . . 5 (𝜑 → (ran 𝐹 × ran 𝐹) = (𝐵 × 𝐵))
3834, 37sseqtrrd 3956 . . . 4 (𝜑 → dom ⊆ (ran 𝐹 × ran 𝐹))
3910eleq2d 2875 . . . . . . . . . . . . 13 (𝜑 → (⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ ↔ ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩}))
4039adantr 484 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → (⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ ↔ ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩}))
41 df-br 5031 . . . . . . . . . . . 12 (⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤 ↔ ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ )
42 eliun 4885 . . . . . . . . . . . . 13 (⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ↔ ∃𝑝𝑉 ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
43 eliun 4885 . . . . . . . . . . . . . 14 (⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ↔ ∃𝑞𝑉 ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
4443rexbii 3210 . . . . . . . . . . . . 13 (∃𝑝𝑉 ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ↔ ∃𝑝𝑉𝑞𝑉 ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
4542, 44bitr2i 279 . . . . . . . . . . . 12 (∃𝑝𝑉𝑞𝑉 ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ↔ ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
4640, 41, 453bitr4g 317 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → (⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤 ↔ ∃𝑝𝑉𝑞𝑉 ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩}))
47 opex 5321 . . . . . . . . . . . . . . 15 ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ V
4847elsn 4540 . . . . . . . . . . . . . 14 (⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ↔ ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ = ⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩)
49 opex 5321 . . . . . . . . . . . . . . . 16 ⟨(𝐹𝑎), (𝐹𝑏)⟩ ∈ V
50 vex 3444 . . . . . . . . . . . . . . . 16 𝑤 ∈ V
5149, 50opth 5333 . . . . . . . . . . . . . . 15 (⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ = ⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ ↔ (⟨(𝐹𝑎), (𝐹𝑏)⟩ = ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∧ 𝑤 = (𝐹‘(𝑝 · 𝑞))))
52 fvex 6658 . . . . . . . . . . . . . . . . . . 19 (𝐹𝑎) ∈ V
53 fvex 6658 . . . . . . . . . . . . . . . . . . 19 (𝐹𝑏) ∈ V
5452, 53opth 5333 . . . . . . . . . . . . . . . . . 18 (⟨(𝐹𝑎), (𝐹𝑏)⟩ = ⟨(𝐹𝑝), (𝐹𝑞)⟩ ↔ ((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)))
55 imasaddf.e . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))
5654, 55syl5bi 245 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (⟨(𝐹𝑎), (𝐹𝑏)⟩ = ⟨(𝐹𝑝), (𝐹𝑞)⟩ → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))
57 eqeq2 2810 . . . . . . . . . . . . . . . . . 18 ((𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞)) → (𝑤 = (𝐹‘(𝑎 · 𝑏)) ↔ 𝑤 = (𝐹‘(𝑝 · 𝑞))))
5857biimprd 251 . . . . . . . . . . . . . . . . 17 ((𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞)) → (𝑤 = (𝐹‘(𝑝 · 𝑞)) → 𝑤 = (𝐹‘(𝑎 · 𝑏))))
5956, 58syl6 35 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (⟨(𝐹𝑎), (𝐹𝑏)⟩ = ⟨(𝐹𝑝), (𝐹𝑞)⟩ → (𝑤 = (𝐹‘(𝑝 · 𝑞)) → 𝑤 = (𝐹‘(𝑎 · 𝑏)))))
6059impd 414 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → ((⟨(𝐹𝑎), (𝐹𝑏)⟩ = ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∧ 𝑤 = (𝐹‘(𝑝 · 𝑞))) → 𝑤 = (𝐹‘(𝑎 · 𝑏))))
6151, 60syl5bi 245 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ = ⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ → 𝑤 = (𝐹‘(𝑎 · 𝑏))))
6248, 61syl5bi 245 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} → 𝑤 = (𝐹‘(𝑎 · 𝑏))))
63623expa 1115 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑝𝑉𝑞𝑉)) → (⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} → 𝑤 = (𝐹‘(𝑎 · 𝑏))))
6463rexlimdvva 3253 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → (∃𝑝𝑉𝑞𝑉 ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} → 𝑤 = (𝐹‘(𝑎 · 𝑏))))
6546, 64sylbid 243 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → (⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤𝑤 = (𝐹‘(𝑎 · 𝑏))))
6665alrimiv 1928 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → ∀𝑤(⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤𝑤 = (𝐹‘(𝑎 · 𝑏))))
67 mo2icl 3653 . . . . . . . . 9 (∀𝑤(⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤𝑤 = (𝐹‘(𝑎 · 𝑏))) → ∃*𝑤⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤)
6866, 67syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → ∃*𝑤⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤)
6968ralrimivva 3156 . . . . . . 7 (𝜑 → ∀𝑎𝑉𝑏𝑉 ∃*𝑤⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤)
70 fofn 6567 . . . . . . . . . 10 (𝐹:𝑉onto𝐵𝐹 Fn 𝑉)
7113, 70syl 17 . . . . . . . . 9 (𝜑𝐹 Fn 𝑉)
72 opeq2 4765 . . . . . . . . . . . 12 (𝑧 = (𝐹𝑏) → ⟨(𝐹𝑎), 𝑧⟩ = ⟨(𝐹𝑎), (𝐹𝑏)⟩)
7372breq1d 5040 . . . . . . . . . . 11 (𝑧 = (𝐹𝑏) → (⟨(𝐹𝑎), 𝑧 𝑤 ↔ ⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤))
7473mobidv 2608 . . . . . . . . . 10 (𝑧 = (𝐹𝑏) → (∃*𝑤⟨(𝐹𝑎), 𝑧 𝑤 ↔ ∃*𝑤⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤))
7574ralrn 6831 . . . . . . . . 9 (𝐹 Fn 𝑉 → (∀𝑧 ∈ ran 𝐹∃*𝑤⟨(𝐹𝑎), 𝑧 𝑤 ↔ ∀𝑏𝑉 ∃*𝑤⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤))
7671, 75syl 17 . . . . . . . 8 (𝜑 → (∀𝑧 ∈ ran 𝐹∃*𝑤⟨(𝐹𝑎), 𝑧 𝑤 ↔ ∀𝑏𝑉 ∃*𝑤⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤))
7776ralbidv 3162 . . . . . . 7 (𝜑 → (∀𝑎𝑉𝑧 ∈ ran 𝐹∃*𝑤⟨(𝐹𝑎), 𝑧 𝑤 ↔ ∀𝑎𝑉𝑏𝑉 ∃*𝑤⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤))
7869, 77mpbird 260 . . . . . 6 (𝜑 → ∀𝑎𝑉𝑧 ∈ ran 𝐹∃*𝑤⟨(𝐹𝑎), 𝑧 𝑤)
79 opeq1 4763 . . . . . . . . . . 11 (𝑦 = (𝐹𝑎) → ⟨𝑦, 𝑧⟩ = ⟨(𝐹𝑎), 𝑧⟩)
8079breq1d 5040 . . . . . . . . . 10 (𝑦 = (𝐹𝑎) → (⟨𝑦, 𝑧 𝑤 ↔ ⟨(𝐹𝑎), 𝑧 𝑤))
8180mobidv 2608 . . . . . . . . 9 (𝑦 = (𝐹𝑎) → (∃*𝑤𝑦, 𝑧 𝑤 ↔ ∃*𝑤⟨(𝐹𝑎), 𝑧 𝑤))
8281ralbidv 3162 . . . . . . . 8 (𝑦 = (𝐹𝑎) → (∀𝑧 ∈ ran 𝐹∃*𝑤𝑦, 𝑧 𝑤 ↔ ∀𝑧 ∈ ran 𝐹∃*𝑤⟨(𝐹𝑎), 𝑧 𝑤))
8382ralrn 6831 . . . . . . 7 (𝐹 Fn 𝑉 → (∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐹∃*𝑤𝑦, 𝑧 𝑤 ↔ ∀𝑎𝑉𝑧 ∈ ran 𝐹∃*𝑤⟨(𝐹𝑎), 𝑧 𝑤))
8471, 83syl 17 . . . . . 6 (𝜑 → (∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐹∃*𝑤𝑦, 𝑧 𝑤 ↔ ∀𝑎𝑉𝑧 ∈ ran 𝐹∃*𝑤⟨(𝐹𝑎), 𝑧 𝑤))
8578, 84mpbird 260 . . . . 5 (𝜑 → ∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐹∃*𝑤𝑦, 𝑧 𝑤)
86 breq1 5033 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 𝑤 ↔ ⟨𝑦, 𝑧 𝑤))
8786mobidv 2608 . . . . . 6 (𝑥 = ⟨𝑦, 𝑧⟩ → (∃*𝑤 𝑥 𝑤 ↔ ∃*𝑤𝑦, 𝑧 𝑤))
8887ralxp 5676 . . . . 5 (∀𝑥 ∈ (ran 𝐹 × ran 𝐹)∃*𝑤 𝑥 𝑤 ↔ ∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐹∃*𝑤𝑦, 𝑧 𝑤)
8985, 88sylibr 237 . . . 4 (𝜑 → ∀𝑥 ∈ (ran 𝐹 × ran 𝐹)∃*𝑤 𝑥 𝑤)
90 ssralv 3981 . . . 4 (dom ⊆ (ran 𝐹 × ran 𝐹) → (∀𝑥 ∈ (ran 𝐹 × ran 𝐹)∃*𝑤 𝑥 𝑤 → ∀𝑥 ∈ dom ∃*𝑤 𝑥 𝑤))
9138, 89, 90sylc 65 . . 3 (𝜑 → ∀𝑥 ∈ dom ∃*𝑤 𝑥 𝑤)
92 dffun7 6351 . . 3 (Fun ↔ (Rel ∧ ∀𝑥 ∈ dom ∃*𝑤 𝑥 𝑤))
9312, 91, 92sylanbrc 586 . 2 (𝜑 → Fun )
94 eqimss2 3972 . . . . . . . . . . 11 ( = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} → 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ )
9510, 94syl 17 . . . . . . . . . 10 (𝜑 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ )
96 iunss 4932 . . . . . . . . . 10 ( 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ↔ ∀𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ )
9795, 96sylib 221 . . . . . . . . 9 (𝜑 → ∀𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ )
98 iunss 4932 . . . . . . . . . . 11 ( 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ↔ ∀𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ )
99 opex 5321 . . . . . . . . . . . . . 14 ⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ ∈ V
10099snss 4679 . . . . . . . . . . . . 13 (⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ ∈ ↔ {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ )
1011, 2opeldm 5740 . . . . . . . . . . . . 13 (⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ ∈ → ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ dom )
102100, 101sylbir 238 . . . . . . . . . . . 12 ({⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ → ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ dom )
103102ralimi 3128 . . . . . . . . . . 11 (∀𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ → ∀𝑞𝑉 ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ dom )
10498, 103sylbi 220 . . . . . . . . . 10 ( 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ → ∀𝑞𝑉 ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ dom )
105104ralimi 3128 . . . . . . . . 9 (∀𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ → ∀𝑝𝑉𝑞𝑉 ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ dom )
10697, 105syl 17 . . . . . . . 8 (𝜑 → ∀𝑝𝑉𝑞𝑉 ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ dom )
107 opeq2 4765 . . . . . . . . . . . 12 (𝑧 = (𝐹𝑞) → ⟨(𝐹𝑝), 𝑧⟩ = ⟨(𝐹𝑝), (𝐹𝑞)⟩)
108107eleq1d 2874 . . . . . . . . . . 11 (𝑧 = (𝐹𝑞) → (⟨(𝐹𝑝), 𝑧⟩ ∈ dom ↔ ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ dom ))
109108ralrn 6831 . . . . . . . . . 10 (𝐹 Fn 𝑉 → (∀𝑧 ∈ ran 𝐹⟨(𝐹𝑝), 𝑧⟩ ∈ dom ↔ ∀𝑞𝑉 ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ dom ))
11071, 109syl 17 . . . . . . . . 9 (𝜑 → (∀𝑧 ∈ ran 𝐹⟨(𝐹𝑝), 𝑧⟩ ∈ dom ↔ ∀𝑞𝑉 ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ dom ))
111110ralbidv 3162 . . . . . . . 8 (𝜑 → (∀𝑝𝑉𝑧 ∈ ran 𝐹⟨(𝐹𝑝), 𝑧⟩ ∈ dom ↔ ∀𝑝𝑉𝑞𝑉 ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ dom ))
112106, 111mpbird 260 . . . . . . 7 (𝜑 → ∀𝑝𝑉𝑧 ∈ ran 𝐹⟨(𝐹𝑝), 𝑧⟩ ∈ dom )
113 opeq1 4763 . . . . . . . . . . 11 (𝑦 = (𝐹𝑝) → ⟨𝑦, 𝑧⟩ = ⟨(𝐹𝑝), 𝑧⟩)
114113eleq1d 2874 . . . . . . . . . 10 (𝑦 = (𝐹𝑝) → (⟨𝑦, 𝑧⟩ ∈ dom ↔ ⟨(𝐹𝑝), 𝑧⟩ ∈ dom ))
115114ralbidv 3162 . . . . . . . . 9 (𝑦 = (𝐹𝑝) → (∀𝑧 ∈ ran 𝐹𝑦, 𝑧⟩ ∈ dom ↔ ∀𝑧 ∈ ran 𝐹⟨(𝐹𝑝), 𝑧⟩ ∈ dom ))
116115ralrn 6831 . . . . . . . 8 (𝐹 Fn 𝑉 → (∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐹𝑦, 𝑧⟩ ∈ dom ↔ ∀𝑝𝑉𝑧 ∈ ran 𝐹⟨(𝐹𝑝), 𝑧⟩ ∈ dom ))
11771, 116syl 17 . . . . . . 7 (𝜑 → (∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐹𝑦, 𝑧⟩ ∈ dom ↔ ∀𝑝𝑉𝑧 ∈ ran 𝐹⟨(𝐹𝑝), 𝑧⟩ ∈ dom ))
118112, 117mpbird 260 . . . . . 6 (𝜑 → ∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐹𝑦, 𝑧⟩ ∈ dom )
119 eleq1 2877 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ dom ↔ ⟨𝑦, 𝑧⟩ ∈ dom ))
120119ralxp 5676 . . . . . 6 (∀𝑥 ∈ (ran 𝐹 × ran 𝐹)𝑥 ∈ dom ↔ ∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐹𝑦, 𝑧⟩ ∈ dom )
121118, 120sylibr 237 . . . . 5 (𝜑 → ∀𝑥 ∈ (ran 𝐹 × ran 𝐹)𝑥 ∈ dom )
122 dfss3 3903 . . . . 5 ((ran 𝐹 × ran 𝐹) ⊆ dom ↔ ∀𝑥 ∈ (ran 𝐹 × ran 𝐹)𝑥 ∈ dom )
123121, 122sylibr 237 . . . 4 (𝜑 → (ran 𝐹 × ran 𝐹) ⊆ dom )
12437, 123eqsstrrd 3954 . . 3 (𝜑 → (𝐵 × 𝐵) ⊆ dom )
12534, 124eqssd 3932 . 2 (𝜑 → dom = (𝐵 × 𝐵))
126 df-fn 6327 . 2 ( Fn (𝐵 × 𝐵) ↔ (Fun ∧ dom = (𝐵 × 𝐵)))
12793, 125, 126sylanbrc 586 1 (𝜑 Fn (𝐵 × 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084  ∀wal 1536   = wceq 1538   ∈ wcel 2111  ∃*wmo 2596   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  {csn 4525  ⟨cop 4531  ∪ ciun 4881   class class class wbr 5030   × cxp 5517  dom cdm 5519  ran crn 5520  Rel wrel 5524  Fun wfun 6318   Fn wfn 6319  ⟶wf 6320  –onto→wfo 6322  ‘cfv 6324  (class class class)co 7135 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fo 6330  df-fv 6332 This theorem is referenced by:  imasaddvallem  16796  imasaddflem  16797  imasaddfn  16798  imasmulfn  16801
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