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Theorem imasaddfnlem 17410
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.f (𝜑𝐹:𝑉onto𝐵)
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)

Proof of Theorem imasaddfnlem
Dummy variables 𝑤 𝑦 𝑧 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opex 5421 . . . . . . . . 9 ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ V
2 fvex 6855 . . . . . . . . 9 (𝐹‘(𝑝 · 𝑞)) ∈ V
31, 2relsnop 5761 . . . . . . . 8 Rel {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩}
43rgenw 3068 . . . . . . 7 𝑞𝑉 Rel {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩}
5 reliun 5772 . . . . . . 7 (Rel 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ↔ ∀𝑞𝑉 Rel {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
64, 5mpbir 230 . . . . . 6 Rel 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩}
76rgenw 3068 . . . . 5 𝑝𝑉 Rel 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩}
8 reliun 5772 . . . . 5 (Rel 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ↔ ∀𝑝𝑉 Rel 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
97, 8mpbir 230 . . . 4 Rel 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩}
10 imasaddflem.a . . . . 5 (𝜑 = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
1110releqd 5734 . . . 4 (𝜑 → (Rel ↔ Rel 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩}))
129, 11mpbiri 257 . . 3 (𝜑 → Rel )
13 imasaddf.f . . . . . . . . . . . . . . . 16 (𝜑𝐹:𝑉onto𝐵)
14 fof 6756 . . . . . . . . . . . . . . . 16 (𝐹:𝑉onto𝐵𝐹:𝑉𝐵)
1513, 14syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐹:𝑉𝐵)
16 ffvelcdm 7032 . . . . . . . . . . . . . . . 16 ((𝐹:𝑉𝐵𝑝𝑉) → (𝐹𝑝) ∈ 𝐵)
17 ffvelcdm 7032 . . . . . . . . . . . . . . . 16 ((𝐹:𝑉𝐵𝑞𝑉) → (𝐹𝑞) ∈ 𝐵)
1816, 17anim12dan 619 . . . . . . . . . . . . . . 15 ((𝐹:𝑉𝐵 ∧ (𝑝𝑉𝑞𝑉)) → ((𝐹𝑝) ∈ 𝐵 ∧ (𝐹𝑞) ∈ 𝐵))
1915, 18sylan 580 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → ((𝐹𝑝) ∈ 𝐵 ∧ (𝐹𝑞) ∈ 𝐵))
20 opelxpi 5670 . . . . . . . . . . . . . 14 (((𝐹𝑝) ∈ 𝐵 ∧ (𝐹𝑞) ∈ 𝐵) → ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ (𝐵 × 𝐵))
2119, 20syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ (𝐵 × 𝐵))
22 opelxpi 5670 . . . . . . . . . . . . 13 ((⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ (𝐵 × 𝐵) ∧ (𝐹‘(𝑝 · 𝑞)) ∈ V) → ⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ ∈ ((𝐵 × 𝐵) × V))
2321, 2, 22sylancl 586 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → ⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ ∈ ((𝐵 × 𝐵) × V))
2423snssd 4769 . . . . . . . . . . 11 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × V))
2524anassrs 468 . . . . . . . . . 10 (((𝜑𝑝𝑉) ∧ 𝑞𝑉) → {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × V))
2625iunssd 5010 . . . . . . . . 9 ((𝜑𝑝𝑉) → 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × V))
2726iunssd 5010 . . . . . . . 8 (𝜑 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × V))
2810, 27eqsstrd 3982 . . . . . . 7 (𝜑 ⊆ ((𝐵 × 𝐵) × V))
29 dmss 5858 . . . . . . 7 ( ⊆ ((𝐵 × 𝐵) × V) → dom ⊆ dom ((𝐵 × 𝐵) × V))
3028, 29syl 17 . . . . . 6 (𝜑 → dom ⊆ dom ((𝐵 × 𝐵) × V))
31 vn0 4298 . . . . . . 7 V ≠ ∅
32 dmxp 5884 . . . . . . 7 (V ≠ ∅ → dom ((𝐵 × 𝐵) × V) = (𝐵 × 𝐵))
3331, 32ax-mp 5 . . . . . 6 dom ((𝐵 × 𝐵) × V) = (𝐵 × 𝐵)
3430, 33sseqtrdi 3994 . . . . 5 (𝜑 → dom ⊆ (𝐵 × 𝐵))
35 forn 6759 . . . . . . 7 (𝐹:𝑉onto𝐵 → ran 𝐹 = 𝐵)
3613, 35syl 17 . . . . . 6 (𝜑 → ran 𝐹 = 𝐵)
3736sqxpeqd 5665 . . . . 5 (𝜑 → (ran 𝐹 × ran 𝐹) = (𝐵 × 𝐵))
3834, 37sseqtrrd 3985 . . . 4 (𝜑 → dom ⊆ (ran 𝐹 × ran 𝐹))
3910eleq2d 2823 . . . . . . . . . . . . 13 (𝜑 → (⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ ↔ ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩}))
4039adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → (⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ ↔ ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩}))
41 df-br 5106 . . . . . . . . . . . 12 (⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤 ↔ ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ )
42 eliun 4958 . . . . . . . . . . . . 13 (⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ↔ ∃𝑝𝑉 ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
43 eliun 4958 . . . . . . . . . . . . . 14 (⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ↔ ∃𝑞𝑉 ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
4443rexbii 3097 . . . . . . . . . . . . 13 (∃𝑝𝑉 ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ↔ ∃𝑝𝑉𝑞𝑉 ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
4542, 44bitr2i 275 . . . . . . . . . . . 12 (∃𝑝𝑉𝑞𝑉 ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ↔ ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
4640, 41, 453bitr4g 313 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → (⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤 ↔ ∃𝑝𝑉𝑞𝑉 ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩}))
47 opex 5421 . . . . . . . . . . . . . . 15 ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ V
4847elsn 4601 . . . . . . . . . . . . . 14 (⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ↔ ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ = ⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩)
49 opex 5421 . . . . . . . . . . . . . . . 16 ⟨(𝐹𝑎), (𝐹𝑏)⟩ ∈ V
50 vex 3449 . . . . . . . . . . . . . . . 16 𝑤 ∈ V
5149, 50opth 5433 . . . . . . . . . . . . . . 15 (⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ = ⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ ↔ (⟨(𝐹𝑎), (𝐹𝑏)⟩ = ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∧ 𝑤 = (𝐹‘(𝑝 · 𝑞))))
52 fvex 6855 . . . . . . . . . . . . . . . . . . 19 (𝐹𝑎) ∈ V
53 fvex 6855 . . . . . . . . . . . . . . . . . . 19 (𝐹𝑏) ∈ V
5452, 53opth 5433 . . . . . . . . . . . . . . . . . 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, 55biimtrid 241 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (⟨(𝐹𝑎), (𝐹𝑏)⟩ = ⟨(𝐹𝑝), (𝐹𝑞)⟩ → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))
57 eqeq2 2748 . . . . . . . . . . . . . . . . . 18 ((𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞)) → (𝑤 = (𝐹‘(𝑎 · 𝑏)) ↔ 𝑤 = (𝐹‘(𝑝 · 𝑞))))
5857biimprd 247 . . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → ((⟨(𝐹𝑎), (𝐹𝑏)⟩ = ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∧ 𝑤 = (𝐹‘(𝑝 · 𝑞))) → 𝑤 = (𝐹‘(𝑎 · 𝑏))))
6151, 60biimtrid 241 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ = ⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ → 𝑤 = (𝐹‘(𝑎 · 𝑏))))
6248, 61biimtrid 241 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} → 𝑤 = (𝐹‘(𝑎 · 𝑏))))
63623expa 1118 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑝𝑉𝑞𝑉)) → (⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} → 𝑤 = (𝐹‘(𝑎 · 𝑏))))
6463rexlimdvva 3205 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → (∃𝑝𝑉𝑞𝑉 ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} → 𝑤 = (𝐹‘(𝑎 · 𝑏))))
6546, 64sylbid 239 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → (⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤𝑤 = (𝐹‘(𝑎 · 𝑏))))
6665alrimiv 1930 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → ∀𝑤(⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤𝑤 = (𝐹‘(𝑎 · 𝑏))))
67 mo2icl 3672 . . . . . . . . 9 (∀𝑤(⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤𝑤 = (𝐹‘(𝑎 · 𝑏))) → ∃*𝑤⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤)
6866, 67syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → ∃*𝑤⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤)
6968ralrimivva 3197 . . . . . . 7 (𝜑 → ∀𝑎𝑉𝑏𝑉 ∃*𝑤⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤)
70 fofn 6758 . . . . . . . . . 10 (𝐹:𝑉onto𝐵𝐹 Fn 𝑉)
7113, 70syl 17 . . . . . . . . 9 (𝜑𝐹 Fn 𝑉)
72 opeq2 4831 . . . . . . . . . . . 12 (𝑧 = (𝐹𝑏) → ⟨(𝐹𝑎), 𝑧⟩ = ⟨(𝐹𝑎), (𝐹𝑏)⟩)
7372breq1d 5115 . . . . . . . . . . 11 (𝑧 = (𝐹𝑏) → (⟨(𝐹𝑎), 𝑧 𝑤 ↔ ⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤))
7473mobidv 2547 . . . . . . . . . 10 (𝑧 = (𝐹𝑏) → (∃*𝑤⟨(𝐹𝑎), 𝑧 𝑤 ↔ ∃*𝑤⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤))
7574ralrn 7038 . . . . . . . . 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 3174 . . . . . . 7 (𝜑 → (∀𝑎𝑉𝑧 ∈ ran 𝐹∃*𝑤⟨(𝐹𝑎), 𝑧 𝑤 ↔ ∀𝑎𝑉𝑏𝑉 ∃*𝑤⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤))
7869, 77mpbird 256 . . . . . 6 (𝜑 → ∀𝑎𝑉𝑧 ∈ ran 𝐹∃*𝑤⟨(𝐹𝑎), 𝑧 𝑤)
79 opeq1 4830 . . . . . . . . . . 11 (𝑦 = (𝐹𝑎) → ⟨𝑦, 𝑧⟩ = ⟨(𝐹𝑎), 𝑧⟩)
8079breq1d 5115 . . . . . . . . . 10 (𝑦 = (𝐹𝑎) → (⟨𝑦, 𝑧 𝑤 ↔ ⟨(𝐹𝑎), 𝑧 𝑤))
8180mobidv 2547 . . . . . . . . 9 (𝑦 = (𝐹𝑎) → (∃*𝑤𝑦, 𝑧 𝑤 ↔ ∃*𝑤⟨(𝐹𝑎), 𝑧 𝑤))
8281ralbidv 3174 . . . . . . . 8 (𝑦 = (𝐹𝑎) → (∀𝑧 ∈ ran 𝐹∃*𝑤𝑦, 𝑧 𝑤 ↔ ∀𝑧 ∈ ran 𝐹∃*𝑤⟨(𝐹𝑎), 𝑧 𝑤))
8382ralrn 7038 . . . . . . 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 256 . . . . 5 (𝜑 → ∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐹∃*𝑤𝑦, 𝑧 𝑤)
86 breq1 5108 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 𝑤 ↔ ⟨𝑦, 𝑧 𝑤))
8786mobidv 2547 . . . . . 6 (𝑥 = ⟨𝑦, 𝑧⟩ → (∃*𝑤 𝑥 𝑤 ↔ ∃*𝑤𝑦, 𝑧 𝑤))
8887ralxp 5797 . . . . 5 (∀𝑥 ∈ (ran 𝐹 × ran 𝐹)∃*𝑤 𝑥 𝑤 ↔ ∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐹∃*𝑤𝑦, 𝑧 𝑤)
8985, 88sylibr 233 . . . 4 (𝜑 → ∀𝑥 ∈ (ran 𝐹 × ran 𝐹)∃*𝑤 𝑥 𝑤)
90 ssralv 4010 . . . 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 6528 . . 3 (Fun ↔ (Rel ∧ ∀𝑥 ∈ dom ∃*𝑤 𝑥 𝑤))
9312, 91, 92sylanbrc 583 . 2 (𝜑 → Fun )
94 eqimss2 4001 . . . . . . . . . . 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 5005 . . . . . . . . . 10 ( 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ↔ ∀𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ )
9795, 96sylib 217 . . . . . . . . 9 (𝜑 → ∀𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ )
98 iunss 5005 . . . . . . . . . . 11 ( 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ↔ ∀𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ )
99 opex 5421 . . . . . . . . . . . . . 14 ⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ ∈ V
10099snss 4746 . . . . . . . . . . . . 13 (⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ ∈ ↔ {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ )
1011, 2opeldm 5863 . . . . . . . . . . . . 13 (⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ ∈ → ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ dom )
102100, 101sylbir 234 . . . . . . . . . . . 12 ({⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ → ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ dom )
103102ralimi 3086 . . . . . . . . . . 11 (∀𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ → ∀𝑞𝑉 ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ dom )
10498, 103sylbi 216 . . . . . . . . . 10 ( 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ → ∀𝑞𝑉 ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ dom )
105104ralimi 3086 . . . . . . . . 9 (∀𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ → ∀𝑝𝑉𝑞𝑉 ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ dom )
10697, 105syl 17 . . . . . . . 8 (𝜑 → ∀𝑝𝑉𝑞𝑉 ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ dom )
107 opeq2 4831 . . . . . . . . . . . 12 (𝑧 = (𝐹𝑞) → ⟨(𝐹𝑝), 𝑧⟩ = ⟨(𝐹𝑝), (𝐹𝑞)⟩)
108107eleq1d 2822 . . . . . . . . . . 11 (𝑧 = (𝐹𝑞) → (⟨(𝐹𝑝), 𝑧⟩ ∈ dom ↔ ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ dom ))
109108ralrn 7038 . . . . . . . . . 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 3174 . . . . . . . 8 (𝜑 → (∀𝑝𝑉𝑧 ∈ ran 𝐹⟨(𝐹𝑝), 𝑧⟩ ∈ dom ↔ ∀𝑝𝑉𝑞𝑉 ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ dom ))
112106, 111mpbird 256 . . . . . . 7 (𝜑 → ∀𝑝𝑉𝑧 ∈ ran 𝐹⟨(𝐹𝑝), 𝑧⟩ ∈ dom )
113 opeq1 4830 . . . . . . . . . . 11 (𝑦 = (𝐹𝑝) → ⟨𝑦, 𝑧⟩ = ⟨(𝐹𝑝), 𝑧⟩)
114113eleq1d 2822 . . . . . . . . . 10 (𝑦 = (𝐹𝑝) → (⟨𝑦, 𝑧⟩ ∈ dom ↔ ⟨(𝐹𝑝), 𝑧⟩ ∈ dom ))
115114ralbidv 3174 . . . . . . . . 9 (𝑦 = (𝐹𝑝) → (∀𝑧 ∈ ran 𝐹𝑦, 𝑧⟩ ∈ dom ↔ ∀𝑧 ∈ ran 𝐹⟨(𝐹𝑝), 𝑧⟩ ∈ dom ))
116115ralrn 7038 . . . . . . . 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 256 . . . . . 6 (𝜑 → ∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐹𝑦, 𝑧⟩ ∈ dom )
119 eleq1 2825 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ dom ↔ ⟨𝑦, 𝑧⟩ ∈ dom ))
120119ralxp 5797 . . . . . 6 (∀𝑥 ∈ (ran 𝐹 × ran 𝐹)𝑥 ∈ dom ↔ ∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐹𝑦, 𝑧⟩ ∈ dom )
121118, 120sylibr 233 . . . . 5 (𝜑 → ∀𝑥 ∈ (ran 𝐹 × ran 𝐹)𝑥 ∈ dom )
122 dfss3 3932 . . . . 5 ((ran 𝐹 × ran 𝐹) ⊆ dom ↔ ∀𝑥 ∈ (ran 𝐹 × ran 𝐹)𝑥 ∈ dom )
123121, 122sylibr 233 . . . 4 (𝜑 → (ran 𝐹 × ran 𝐹) ⊆ dom )
12437, 123eqsstrrd 3983 . . 3 (𝜑 → (𝐵 × 𝐵) ⊆ dom )
12534, 124eqssd 3961 . 2 (𝜑 → dom = (𝐵 × 𝐵))
126 df-fn 6499 . 2 ( Fn (𝐵 × 𝐵) ↔ (Fun ∧ dom = (𝐵 × 𝐵)))
12793, 125, 126sylanbrc 583 1 (𝜑 Fn (𝐵 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087  wal 1539   = wceq 1541  wcel 2106  ∃*wmo 2536  wne 2943  wral 3064  wrex 3073  Vcvv 3445  wss 3910  c0 4282  {csn 4586  cop 4592   ciun 4954   class class class wbr 5105   × cxp 5631  dom cdm 5633  ran crn 5634  Rel wrel 5638  Fun wfun 6490   Fn wfn 6491  wf 6492  ontowfo 6494  cfv 6496  (class class class)co 7357
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fo 6502  df-fv 6504
This theorem is referenced by:  imasaddvallem  17411  imasaddflem  17412  imasaddfn  17413  imasmulfn  17416
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