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Theorem imasaddflem 16384
Description: The image set operations are closed if the original operation is. (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))⟩})
imasaddflem.c ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)
Assertion
Ref Expression
imasaddflem (𝜑 :(𝐵 × 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 imasaddflem
StepHypRef Expression
1 imasaddf.f . . 3 (𝜑𝐹:𝑉onto𝐵)
2 imasaddf.e . . 3 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))
3 imasaddflem.a . . 3 (𝜑 = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
41, 2, 3imasaddfnlem 16382 . 2 (𝜑 Fn (𝐵 × 𝐵))
5 fof 6268 . . . . . . . . . . . 12 (𝐹:𝑉onto𝐵𝐹:𝑉𝐵)
61, 5syl 17 . . . . . . . . . . 11 (𝜑𝐹:𝑉𝐵)
7 ffvelrn 6512 . . . . . . . . . . . . 13 ((𝐹:𝑉𝐵𝑝𝑉) → (𝐹𝑝) ∈ 𝐵)
8 ffvelrn 6512 . . . . . . . . . . . . 13 ((𝐹:𝑉𝐵𝑞𝑉) → (𝐹𝑞) ∈ 𝐵)
97, 8anim12dan 918 . . . . . . . . . . . 12 ((𝐹:𝑉𝐵 ∧ (𝑝𝑉𝑞𝑉)) → ((𝐹𝑝) ∈ 𝐵 ∧ (𝐹𝑞) ∈ 𝐵))
10 opelxpi 5297 . . . . . . . . . . . 12 (((𝐹𝑝) ∈ 𝐵 ∧ (𝐹𝑞) ∈ 𝐵) → ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ (𝐵 × 𝐵))
119, 10syl 17 . . . . . . . . . . 11 ((𝐹:𝑉𝐵 ∧ (𝑝𝑉𝑞𝑉)) → ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ (𝐵 × 𝐵))
126, 11sylan 489 . . . . . . . . . 10 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ (𝐵 × 𝐵))
13 imasaddflem.c . . . . . . . . . . 11 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)
14 ffvelrn 6512 . . . . . . . . . . . 12 ((𝐹:𝑉𝐵 ∧ (𝑝 · 𝑞) ∈ 𝑉) → (𝐹‘(𝑝 · 𝑞)) ∈ 𝐵)
156, 14sylan 489 . . . . . . . . . . 11 ((𝜑 ∧ (𝑝 · 𝑞) ∈ 𝑉) → (𝐹‘(𝑝 · 𝑞)) ∈ 𝐵)
1613, 15syldan 488 . . . . . . . . . 10 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝐹‘(𝑝 · 𝑞)) ∈ 𝐵)
17 opelxpi 5297 . . . . . . . . . 10 ((⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ (𝐵 × 𝐵) ∧ (𝐹‘(𝑝 · 𝑞)) ∈ 𝐵) → ⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ ∈ ((𝐵 × 𝐵) × 𝐵))
1812, 16, 17syl2anc 696 . . . . . . . . 9 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → ⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ ∈ ((𝐵 × 𝐵) × 𝐵))
1918snssd 4477 . . . . . . . 8 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × 𝐵))
2019anassrs 683 . . . . . . 7 (((𝜑𝑝𝑉) ∧ 𝑞𝑉) → {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × 𝐵))
2120ralrimiva 3096 . . . . . 6 ((𝜑𝑝𝑉) → ∀𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × 𝐵))
22 iunss 4705 . . . . . 6 ( 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × 𝐵) ↔ ∀𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × 𝐵))
2321, 22sylibr 224 . . . . 5 ((𝜑𝑝𝑉) → 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × 𝐵))
2423ralrimiva 3096 . . . 4 (𝜑 → ∀𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × 𝐵))
25 iunss 4705 . . . 4 ( 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × 𝐵) ↔ ∀𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × 𝐵))
2624, 25sylibr 224 . . 3 (𝜑 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × 𝐵))
273, 26eqsstrd 3772 . 2 (𝜑 ⊆ ((𝐵 × 𝐵) × 𝐵))
28 dff2 6526 . 2 ( :(𝐵 × 𝐵)⟶𝐵 ↔ ( Fn (𝐵 × 𝐵) ∧ ⊆ ((𝐵 × 𝐵) × 𝐵)))
294, 27, 28sylanbrc 701 1 (𝜑 :(𝐵 × 𝐵)⟶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1624  wcel 2131  wral 3042  wss 3707  {csn 4313  cop 4319   ciun 4664   × cxp 5256   Fn wfn 6036  wf 6037  ontowfo 6039  cfv 6041  (class class class)co 6805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-fo 6047  df-fv 6049
This theorem is referenced by:  imasaddf  16387  imasmulf  16390  qusaddflem  16406
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