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Theorem imasaddfnlem 16169
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 4923 . . . . . . . . 9 ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ V
2 fvex 6188 . . . . . . . . 9 (𝐹‘(𝑝 · 𝑞)) ∈ V
31, 2relsnop 5214 . . . . . . . 8 Rel {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩}
43rgenw 2921 . . . . . . 7 𝑞𝑉 Rel {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩}
5 reliun 5229 . . . . . . 7 (Rel 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ↔ ∀𝑞𝑉 Rel {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
64, 5mpbir 221 . . . . . 6 Rel 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩}
76rgenw 2921 . . . . 5 𝑝𝑉 Rel 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩}
8 reliun 5229 . . . . 5 (Rel 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ↔ ∀𝑝𝑉 Rel 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
97, 8mpbir 221 . . . 4 Rel 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩}
10 imasaddflem.a . . . . 5 (𝜑 = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
1110releqd 5193 . . . 4 (𝜑 → (Rel ↔ Rel 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩}))
129, 11mpbiri 248 . . 3 (𝜑 → Rel )
13 imasaddf.f . . . . . . . . . . . . . . . . . 18 (𝜑𝐹:𝑉onto𝐵)
14 fof 6102 . . . . . . . . . . . . . . . . . 18 (𝐹:𝑉onto𝐵𝐹:𝑉𝐵)
1513, 14syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝐹:𝑉𝐵)
16 ffvelrn 6343 . . . . . . . . . . . . . . . . . 18 ((𝐹:𝑉𝐵𝑝𝑉) → (𝐹𝑝) ∈ 𝐵)
17 ffvelrn 6343 . . . . . . . . . . . . . . . . . 18 ((𝐹:𝑉𝐵𝑞𝑉) → (𝐹𝑞) ∈ 𝐵)
1816, 17anim12dan 881 . . . . . . . . . . . . . . . . 17 ((𝐹:𝑉𝐵 ∧ (𝑝𝑉𝑞𝑉)) → ((𝐹𝑝) ∈ 𝐵 ∧ (𝐹𝑞) ∈ 𝐵))
1915, 18sylan 488 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → ((𝐹𝑝) ∈ 𝐵 ∧ (𝐹𝑞) ∈ 𝐵))
20 opelxpi 5138 . . . . . . . . . . . . . . . 16 (((𝐹𝑝) ∈ 𝐵 ∧ (𝐹𝑞) ∈ 𝐵) → ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ (𝐵 × 𝐵))
2119, 20syl 17 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ (𝐵 × 𝐵))
22 opelxpi 5138 . . . . . . . . . . . . . . 15 ((⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ (𝐵 × 𝐵) ∧ (𝐹‘(𝑝 · 𝑞)) ∈ V) → ⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ ∈ ((𝐵 × 𝐵) × V))
2321, 2, 22sylancl 693 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → ⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ ∈ ((𝐵 × 𝐵) × V))
2423snssd 4331 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × V))
2524anassrs 679 . . . . . . . . . . . 12 (((𝜑𝑝𝑉) ∧ 𝑞𝑉) → {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × V))
2625ralrimiva 2963 . . . . . . . . . . 11 ((𝜑𝑝𝑉) → ∀𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × V))
27 iunss 4552 . . . . . . . . . . 11 ( 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × V) ↔ ∀𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × V))
2826, 27sylibr 224 . . . . . . . . . 10 ((𝜑𝑝𝑉) → 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × V))
2928ralrimiva 2963 . . . . . . . . 9 (𝜑 → ∀𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × V))
30 iunss 4552 . . . . . . . . 9 ( 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × V) ↔ ∀𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × V))
3129, 30sylibr 224 . . . . . . . 8 (𝜑 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × V))
3210, 31eqsstrd 3631 . . . . . . 7 (𝜑 ⊆ ((𝐵 × 𝐵) × V))
33 dmss 5312 . . . . . . 7 ( ⊆ ((𝐵 × 𝐵) × V) → dom ⊆ dom ((𝐵 × 𝐵) × V))
3432, 33syl 17 . . . . . 6 (𝜑 → dom ⊆ dom ((𝐵 × 𝐵) × V))
35 vn0 3916 . . . . . . 7 V ≠ ∅
36 dmxp 5333 . . . . . . 7 (V ≠ ∅ → dom ((𝐵 × 𝐵) × V) = (𝐵 × 𝐵))
3735, 36ax-mp 5 . . . . . 6 dom ((𝐵 × 𝐵) × V) = (𝐵 × 𝐵)
3834, 37syl6sseq 3643 . . . . 5 (𝜑 → dom ⊆ (𝐵 × 𝐵))
39 forn 6105 . . . . . . 7 (𝐹:𝑉onto𝐵 → ran 𝐹 = 𝐵)
4013, 39syl 17 . . . . . 6 (𝜑 → ran 𝐹 = 𝐵)
4140sqxpeqd 5131 . . . . 5 (𝜑 → (ran 𝐹 × ran 𝐹) = (𝐵 × 𝐵))
4238, 41sseqtr4d 3634 . . . 4 (𝜑 → dom ⊆ (ran 𝐹 × ran 𝐹))
4310eleq2d 2685 . . . . . . . . . . . . 13 (𝜑 → (⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ ↔ ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩}))
4443adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → (⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ ↔ ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩}))
45 df-br 4645 . . . . . . . . . . . 12 (⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤 ↔ ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ )
46 eliun 4515 . . . . . . . . . . . . 13 (⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ↔ ∃𝑝𝑉 ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
47 eliun 4515 . . . . . . . . . . . . . 14 (⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ↔ ∃𝑞𝑉 ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
4847rexbii 3037 . . . . . . . . . . . . 13 (∃𝑝𝑉 ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ↔ ∃𝑝𝑉𝑞𝑉 ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
4946, 48bitr2i 265 . . . . . . . . . . . 12 (∃𝑝𝑉𝑞𝑉 ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ↔ ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
5044, 45, 493bitr4g 303 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → (⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤 ↔ ∃𝑝𝑉𝑞𝑉 ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩}))
51 opex 4923 . . . . . . . . . . . . . . 15 ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ V
5251elsn 4183 . . . . . . . . . . . . . 14 (⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ↔ ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ = ⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩)
53 opex 4923 . . . . . . . . . . . . . . . 16 ⟨(𝐹𝑎), (𝐹𝑏)⟩ ∈ V
54 vex 3198 . . . . . . . . . . . . . . . 16 𝑤 ∈ V
5553, 54opth 4935 . . . . . . . . . . . . . . 15 (⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ = ⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ ↔ (⟨(𝐹𝑎), (𝐹𝑏)⟩ = ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∧ 𝑤 = (𝐹‘(𝑝 · 𝑞))))
56 fvex 6188 . . . . . . . . . . . . . . . . . . 19 (𝐹𝑎) ∈ V
57 fvex 6188 . . . . . . . . . . . . . . . . . . 19 (𝐹𝑏) ∈ V
5856, 57opth 4935 . . . . . . . . . . . . . . . . . 18 (⟨(𝐹𝑎), (𝐹𝑏)⟩ = ⟨(𝐹𝑝), (𝐹𝑞)⟩ ↔ ((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)))
59 imasaddf.e . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))
6058, 59syl5bi 232 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (⟨(𝐹𝑎), (𝐹𝑏)⟩ = ⟨(𝐹𝑝), (𝐹𝑞)⟩ → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))
61 eqeq2 2631 . . . . . . . . . . . . . . . . . 18 ((𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞)) → (𝑤 = (𝐹‘(𝑎 · 𝑏)) ↔ 𝑤 = (𝐹‘(𝑝 · 𝑞))))
6261biimprd 238 . . . . . . . . . . . . . . . . 17 ((𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞)) → (𝑤 = (𝐹‘(𝑝 · 𝑞)) → 𝑤 = (𝐹‘(𝑎 · 𝑏))))
6360, 62syl6 35 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (⟨(𝐹𝑎), (𝐹𝑏)⟩ = ⟨(𝐹𝑝), (𝐹𝑞)⟩ → (𝑤 = (𝐹‘(𝑝 · 𝑞)) → 𝑤 = (𝐹‘(𝑎 · 𝑏)))))
6463impd 447 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → ((⟨(𝐹𝑎), (𝐹𝑏)⟩ = ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∧ 𝑤 = (𝐹‘(𝑝 · 𝑞))) → 𝑤 = (𝐹‘(𝑎 · 𝑏))))
6555, 64syl5bi 232 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ = ⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ → 𝑤 = (𝐹‘(𝑎 · 𝑏))))
6652, 65syl5bi 232 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} → 𝑤 = (𝐹‘(𝑎 · 𝑏))))
67663expa 1263 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑎𝑉𝑏𝑉)) ∧ (𝑝𝑉𝑞𝑉)) → (⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} → 𝑤 = (𝐹‘(𝑎 · 𝑏))))
6867rexlimdvva 3034 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → (∃𝑝𝑉𝑞𝑉 ⟨⟨(𝐹𝑎), (𝐹𝑏)⟩, 𝑤⟩ ∈ {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} → 𝑤 = (𝐹‘(𝑎 · 𝑏))))
6950, 68sylbid 230 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → (⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤𝑤 = (𝐹‘(𝑎 · 𝑏))))
7069alrimiv 1853 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → ∀𝑤(⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤𝑤 = (𝐹‘(𝑎 · 𝑏))))
71 mo2icl 3379 . . . . . . . . 9 (∀𝑤(⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤𝑤 = (𝐹‘(𝑎 · 𝑏))) → ∃*𝑤⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤)
7270, 71syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑎𝑉𝑏𝑉)) → ∃*𝑤⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤)
7372ralrimivva 2968 . . . . . . 7 (𝜑 → ∀𝑎𝑉𝑏𝑉 ∃*𝑤⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤)
74 fofn 6104 . . . . . . . . . 10 (𝐹:𝑉onto𝐵𝐹 Fn 𝑉)
7513, 74syl 17 . . . . . . . . 9 (𝜑𝐹 Fn 𝑉)
76 opeq2 4394 . . . . . . . . . . . 12 (𝑧 = (𝐹𝑏) → ⟨(𝐹𝑎), 𝑧⟩ = ⟨(𝐹𝑎), (𝐹𝑏)⟩)
7776breq1d 4654 . . . . . . . . . . 11 (𝑧 = (𝐹𝑏) → (⟨(𝐹𝑎), 𝑧 𝑤 ↔ ⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤))
7877mobidv 2489 . . . . . . . . . 10 (𝑧 = (𝐹𝑏) → (∃*𝑤⟨(𝐹𝑎), 𝑧 𝑤 ↔ ∃*𝑤⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤))
7978ralrn 6348 . . . . . . . . 9 (𝐹 Fn 𝑉 → (∀𝑧 ∈ ran 𝐹∃*𝑤⟨(𝐹𝑎), 𝑧 𝑤 ↔ ∀𝑏𝑉 ∃*𝑤⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤))
8075, 79syl 17 . . . . . . . 8 (𝜑 → (∀𝑧 ∈ ran 𝐹∃*𝑤⟨(𝐹𝑎), 𝑧 𝑤 ↔ ∀𝑏𝑉 ∃*𝑤⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤))
8180ralbidv 2983 . . . . . . 7 (𝜑 → (∀𝑎𝑉𝑧 ∈ ran 𝐹∃*𝑤⟨(𝐹𝑎), 𝑧 𝑤 ↔ ∀𝑎𝑉𝑏𝑉 ∃*𝑤⟨(𝐹𝑎), (𝐹𝑏)⟩ 𝑤))
8273, 81mpbird 247 . . . . . 6 (𝜑 → ∀𝑎𝑉𝑧 ∈ ran 𝐹∃*𝑤⟨(𝐹𝑎), 𝑧 𝑤)
83 opeq1 4393 . . . . . . . . . . 11 (𝑦 = (𝐹𝑎) → ⟨𝑦, 𝑧⟩ = ⟨(𝐹𝑎), 𝑧⟩)
8483breq1d 4654 . . . . . . . . . 10 (𝑦 = (𝐹𝑎) → (⟨𝑦, 𝑧 𝑤 ↔ ⟨(𝐹𝑎), 𝑧 𝑤))
8584mobidv 2489 . . . . . . . . 9 (𝑦 = (𝐹𝑎) → (∃*𝑤𝑦, 𝑧 𝑤 ↔ ∃*𝑤⟨(𝐹𝑎), 𝑧 𝑤))
8685ralbidv 2983 . . . . . . . 8 (𝑦 = (𝐹𝑎) → (∀𝑧 ∈ ran 𝐹∃*𝑤𝑦, 𝑧 𝑤 ↔ ∀𝑧 ∈ ran 𝐹∃*𝑤⟨(𝐹𝑎), 𝑧 𝑤))
8786ralrn 6348 . . . . . . 7 (𝐹 Fn 𝑉 → (∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐹∃*𝑤𝑦, 𝑧 𝑤 ↔ ∀𝑎𝑉𝑧 ∈ ran 𝐹∃*𝑤⟨(𝐹𝑎), 𝑧 𝑤))
8875, 87syl 17 . . . . . 6 (𝜑 → (∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐹∃*𝑤𝑦, 𝑧 𝑤 ↔ ∀𝑎𝑉𝑧 ∈ ran 𝐹∃*𝑤⟨(𝐹𝑎), 𝑧 𝑤))
8982, 88mpbird 247 . . . . 5 (𝜑 → ∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐹∃*𝑤𝑦, 𝑧 𝑤)
90 breq1 4647 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 𝑤 ↔ ⟨𝑦, 𝑧 𝑤))
9190mobidv 2489 . . . . . 6 (𝑥 = ⟨𝑦, 𝑧⟩ → (∃*𝑤 𝑥 𝑤 ↔ ∃*𝑤𝑦, 𝑧 𝑤))
9291ralxp 5252 . . . . 5 (∀𝑥 ∈ (ran 𝐹 × ran 𝐹)∃*𝑤 𝑥 𝑤 ↔ ∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐹∃*𝑤𝑦, 𝑧 𝑤)
9389, 92sylibr 224 . . . 4 (𝜑 → ∀𝑥 ∈ (ran 𝐹 × ran 𝐹)∃*𝑤 𝑥 𝑤)
94 ssralv 3658 . . . 4 (dom ⊆ (ran 𝐹 × ran 𝐹) → (∀𝑥 ∈ (ran 𝐹 × ran 𝐹)∃*𝑤 𝑥 𝑤 → ∀𝑥 ∈ dom ∃*𝑤 𝑥 𝑤))
9542, 93, 94sylc 65 . . 3 (𝜑 → ∀𝑥 ∈ dom ∃*𝑤 𝑥 𝑤)
96 dffun7 5903 . . 3 (Fun ↔ (Rel ∧ ∀𝑥 ∈ dom ∃*𝑤 𝑥 𝑤))
9712, 95, 96sylanbrc 697 . 2 (𝜑 → Fun )
98 eqimss2 3650 . . . . . . . . . . 11 ( = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} → 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ )
9910, 98syl 17 . . . . . . . . . 10 (𝜑 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ )
100 iunss 4552 . . . . . . . . . 10 ( 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ↔ ∀𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ )
10199, 100sylib 208 . . . . . . . . 9 (𝜑 → ∀𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ )
102 iunss 4552 . . . . . . . . . . 11 ( 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ↔ ∀𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ )
103 opex 4923 . . . . . . . . . . . . . 14 ⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ ∈ V
104103snss 4307 . . . . . . . . . . . . 13 (⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ ∈ ↔ {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ )
1051, 2opeldm 5317 . . . . . . . . . . . . 13 (⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ ∈ → ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ dom )
106104, 105sylbir 225 . . . . . . . . . . . 12 ({⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ → ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ dom )
107106ralimi 2949 . . . . . . . . . . 11 (∀𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ → ∀𝑞𝑉 ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ dom )
108102, 107sylbi 207 . . . . . . . . . 10 ( 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ → ∀𝑞𝑉 ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ dom )
109108ralimi 2949 . . . . . . . . 9 (∀𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ → ∀𝑝𝑉𝑞𝑉 ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ dom )
110101, 109syl 17 . . . . . . . 8 (𝜑 → ∀𝑝𝑉𝑞𝑉 ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ dom )
111 opeq2 4394 . . . . . . . . . . . 12 (𝑧 = (𝐹𝑞) → ⟨(𝐹𝑝), 𝑧⟩ = ⟨(𝐹𝑝), (𝐹𝑞)⟩)
112111eleq1d 2684 . . . . . . . . . . 11 (𝑧 = (𝐹𝑞) → (⟨(𝐹𝑝), 𝑧⟩ ∈ dom ↔ ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ dom ))
113112ralrn 6348 . . . . . . . . . 10 (𝐹 Fn 𝑉 → (∀𝑧 ∈ ran 𝐹⟨(𝐹𝑝), 𝑧⟩ ∈ dom ↔ ∀𝑞𝑉 ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ dom ))
11475, 113syl 17 . . . . . . . . 9 (𝜑 → (∀𝑧 ∈ ran 𝐹⟨(𝐹𝑝), 𝑧⟩ ∈ dom ↔ ∀𝑞𝑉 ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ dom ))
115114ralbidv 2983 . . . . . . . 8 (𝜑 → (∀𝑝𝑉𝑧 ∈ ran 𝐹⟨(𝐹𝑝), 𝑧⟩ ∈ dom ↔ ∀𝑝𝑉𝑞𝑉 ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ dom ))
116110, 115mpbird 247 . . . . . . 7 (𝜑 → ∀𝑝𝑉𝑧 ∈ ran 𝐹⟨(𝐹𝑝), 𝑧⟩ ∈ dom )
117 opeq1 4393 . . . . . . . . . . 11 (𝑦 = (𝐹𝑝) → ⟨𝑦, 𝑧⟩ = ⟨(𝐹𝑝), 𝑧⟩)
118117eleq1d 2684 . . . . . . . . . 10 (𝑦 = (𝐹𝑝) → (⟨𝑦, 𝑧⟩ ∈ dom ↔ ⟨(𝐹𝑝), 𝑧⟩ ∈ dom ))
119118ralbidv 2983 . . . . . . . . 9 (𝑦 = (𝐹𝑝) → (∀𝑧 ∈ ran 𝐹𝑦, 𝑧⟩ ∈ dom ↔ ∀𝑧 ∈ ran 𝐹⟨(𝐹𝑝), 𝑧⟩ ∈ dom ))
120119ralrn 6348 . . . . . . . 8 (𝐹 Fn 𝑉 → (∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐹𝑦, 𝑧⟩ ∈ dom ↔ ∀𝑝𝑉𝑧 ∈ ran 𝐹⟨(𝐹𝑝), 𝑧⟩ ∈ dom ))
12175, 120syl 17 . . . . . . 7 (𝜑 → (∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐹𝑦, 𝑧⟩ ∈ dom ↔ ∀𝑝𝑉𝑧 ∈ ran 𝐹⟨(𝐹𝑝), 𝑧⟩ ∈ dom ))
122116, 121mpbird 247 . . . . . 6 (𝜑 → ∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐹𝑦, 𝑧⟩ ∈ dom )
123 eleq1 2687 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ dom ↔ ⟨𝑦, 𝑧⟩ ∈ dom ))
124123ralxp 5252 . . . . . 6 (∀𝑥 ∈ (ran 𝐹 × ran 𝐹)𝑥 ∈ dom ↔ ∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐹𝑦, 𝑧⟩ ∈ dom )
125122, 124sylibr 224 . . . . 5 (𝜑 → ∀𝑥 ∈ (ran 𝐹 × ran 𝐹)𝑥 ∈ dom )
126 dfss3 3585 . . . . 5 ((ran 𝐹 × ran 𝐹) ⊆ dom ↔ ∀𝑥 ∈ (ran 𝐹 × ran 𝐹)𝑥 ∈ dom )
127125, 126sylibr 224 . . . 4 (𝜑 → (ran 𝐹 × ran 𝐹) ⊆ dom )
12841, 127eqsstr3d 3632 . . 3 (𝜑 → (𝐵 × 𝐵) ⊆ dom )
12938, 128eqssd 3612 . 2 (𝜑 → dom = (𝐵 × 𝐵))
130 df-fn 5879 . 2 ( Fn (𝐵 × 𝐵) ↔ (Fun ∧ dom = (𝐵 × 𝐵)))
13197, 129, 130sylanbrc 697 1 (𝜑 Fn (𝐵 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wal 1479   = wceq 1481  wcel 1988  ∃*wmo 2469  wne 2791  wral 2909  wrex 2910  Vcvv 3195  wss 3567  c0 3907  {csn 4168  cop 4174   ciun 4511   class class class wbr 4644   × cxp 5102  dom cdm 5104  ran crn 5105  Rel wrel 5109  Fun wfun 5870   Fn wfn 5871  wf 5872  ontowfo 5874  cfv 5876  (class class class)co 6635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fo 5882  df-fv 5884
This theorem is referenced by:  imasaddvallem  16170  imasaddflem  16171  imasaddfn  16172  imasmulfn  16175
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