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Theorem imasaddflem 16111
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 16109 . 2 (𝜑 Fn (𝐵 × 𝐵))
5 fof 6072 . . . . . . . . . . . 12 (𝐹:𝑉onto𝐵𝐹:𝑉𝐵)
61, 5syl 17 . . . . . . . . . . 11 (𝜑𝐹:𝑉𝐵)
7 ffvelrn 6313 . . . . . . . . . . . . 13 ((𝐹:𝑉𝐵𝑝𝑉) → (𝐹𝑝) ∈ 𝐵)
8 ffvelrn 6313 . . . . . . . . . . . . 13 ((𝐹:𝑉𝐵𝑞𝑉) → (𝐹𝑞) ∈ 𝐵)
97, 8anim12dan 881 . . . . . . . . . . . 12 ((𝐹:𝑉𝐵 ∧ (𝑝𝑉𝑞𝑉)) → ((𝐹𝑝) ∈ 𝐵 ∧ (𝐹𝑞) ∈ 𝐵))
10 opelxpi 5108 . . . . . . . . . . . 12 (((𝐹𝑝) ∈ 𝐵 ∧ (𝐹𝑞) ∈ 𝐵) → ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ (𝐵 × 𝐵))
119, 10syl 17 . . . . . . . . . . 11 ((𝐹:𝑉𝐵 ∧ (𝑝𝑉𝑞𝑉)) → ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ (𝐵 × 𝐵))
126, 11sylan 488 . . . . . . . . . 10 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → ⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ (𝐵 × 𝐵))
13 imasaddflem.c . . . . . . . . . . 11 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)
14 ffvelrn 6313 . . . . . . . . . . . 12 ((𝐹:𝑉𝐵 ∧ (𝑝 · 𝑞) ∈ 𝑉) → (𝐹‘(𝑝 · 𝑞)) ∈ 𝐵)
156, 14sylan 488 . . . . . . . . . . 11 ((𝜑 ∧ (𝑝 · 𝑞) ∈ 𝑉) → (𝐹‘(𝑝 · 𝑞)) ∈ 𝐵)
1613, 15syldan 487 . . . . . . . . . 10 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝐹‘(𝑝 · 𝑞)) ∈ 𝐵)
17 opelxpi 5108 . . . . . . . . . 10 ((⟨(𝐹𝑝), (𝐹𝑞)⟩ ∈ (𝐵 × 𝐵) ∧ (𝐹‘(𝑝 · 𝑞)) ∈ 𝐵) → ⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ ∈ ((𝐵 × 𝐵) × 𝐵))
1812, 16, 17syl2anc 692 . . . . . . . . 9 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → ⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ ∈ ((𝐵 × 𝐵) × 𝐵))
1918snssd 4309 . . . . . . . 8 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × 𝐵))
2019anassrs 679 . . . . . . 7 (((𝜑𝑝𝑉) ∧ 𝑞𝑉) → {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × 𝐵))
2120ralrimiva 2960 . . . . . 6 ((𝜑𝑝𝑉) → ∀𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × 𝐵))
22 iunss 4527 . . . . . 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 2960 . . . 4 (𝜑 → ∀𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × 𝐵))
25 iunss 4527 . . . 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 3618 . 2 (𝜑 ⊆ ((𝐵 × 𝐵) × 𝐵))
28 dff2 6327 . 2 ( :(𝐵 × 𝐵)⟶𝐵 ↔ ( Fn (𝐵 × 𝐵) ∧ ⊆ ((𝐵 × 𝐵) × 𝐵)))
294, 27, 28sylanbrc 697 1 (𝜑 :(𝐵 × 𝐵)⟶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wss 3555  {csn 4148  cop 4154   ciun 4485   × cxp 5072   Fn wfn 5842  wf 5843  ontowfo 5845  cfv 5847  (class class class)co 6604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fo 5853  df-fv 5855
This theorem is referenced by:  imasaddf  16114  imasmulf  16117  qusaddflem  16133
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