Theorem imasaddflem 17472

Description: The image set operations are closed if the original operation is. (Contributed by Mario Carneiro, 23-Feb-2015.)

Hypotheses

Ref	Expression
imasaddf.f	⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
imasaddf.e	⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))
imasaddflem.a	⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
imasaddflem.c	⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (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

Step	Hyp	Ref	Expression
1		imasaddf.f	. . 3 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
2		imasaddf.e	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))
3		imasaddflem.a	. . 3 ⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
4	1, 2, 3	imasaddfnlem 17470	. 2 ⊢ (𝜑 → ∙ Fn (𝐵 × 𝐵))
5		fof 6802	. . . . . . . . . 10 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹:𝑉⟶𝐵)
6	1, 5	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑉⟶𝐵)
7		ffvelcdm 7080	. . . . . . . . . . 11 ⊢ ((𝐹:𝑉⟶𝐵 ∧ 𝑝 ∈ 𝑉) → (𝐹‘𝑝) ∈ 𝐵)
8		ffvelcdm 7080	. . . . . . . . . . 11 ⊢ ((𝐹:𝑉⟶𝐵 ∧ 𝑞 ∈ 𝑉) → (𝐹‘𝑞) ∈ 𝐵)
9	7, 8	anim12dan 619	. . . . . . . . . 10 ⊢ ((𝐹:𝑉⟶𝐵 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → ((𝐹‘𝑝) ∈ 𝐵 ∧ (𝐹‘𝑞) ∈ 𝐵))
10		opelxpi 5712	. . . . . . . . . 10 ⊢ (((𝐹‘𝑝) ∈ 𝐵 ∧ (𝐹‘𝑞) ∈ 𝐵) → ⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩ ∈ (𝐵 × 𝐵))
11	9, 10	syl 17	. . . . . . . . 9 ⊢ ((𝐹:𝑉⟶𝐵 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → ⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩ ∈ (𝐵 × 𝐵))
12	6, 11	sylan 580	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → ⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩ ∈ (𝐵 × 𝐵))
13		imasaddflem.c	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)
14		ffvelcdm 7080	. . . . . . . . 9 ⊢ ((𝐹:𝑉⟶𝐵 ∧ (𝑝 · 𝑞) ∈ 𝑉) → (𝐹‘(𝑝 · 𝑞)) ∈ 𝐵)
15	6, 13, 14	syl2an2r 683	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝐹‘(𝑝 · 𝑞)) ∈ 𝐵)
16	12, 15	opelxpd 5713	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ ∈ ((𝐵 × 𝐵) × 𝐵))
17	16	snssd 4811	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × 𝐵))
18	17	anassrs 468	. . . . 5 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × 𝐵))
19	18	iunssd 5052	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑉) → ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × 𝐵))
20	19	iunssd 5052	. . 3 ⊢ (𝜑 → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × 𝐵))
21	3, 20	eqsstrd 4019	. 2 ⊢ (𝜑 → ∙ ⊆ ((𝐵 × 𝐵) × 𝐵))
22		dff2 7097	. 2 ⊢ ( ∙ :(𝐵 × 𝐵)⟶𝐵 ↔ ( ∙ Fn (𝐵 × 𝐵) ∧ ∙ ⊆ ((𝐵 × 𝐵) × 𝐵)))
23	4, 21, 22	sylanbrc 583	1 ⊢ (𝜑 → ∙ :(𝐵 × 𝐵)⟶𝐵)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ⊆ wss 3947  {csn 4627  ⟨cop 4633  ∪ ciun 4996   × cxp 5673   Fn wfn 6535  ⟶wf 6536  –onto→wfo 6538  ‘cfv 6540  (class class class)co 7405

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 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fo 6546  df-fv 6548

This theorem is referenced by:  imasaddf  17475  imasmulf  17478  qusaddflem  17494