Theorem imasaddflemg 13546

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))⟩})
imasaddfnlemg.v	⊢ (𝜑 → 𝑉 ∈ 𝑊)
imasaddfnlemg.x	⊢ (𝜑 → · ∈ 𝐶)
imasaddflem.c	⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)

Assertion

Ref	Expression
imasaddflemg	⊢ (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)   𝐶(𝑞,𝑝,𝑎,𝑏)   · (𝑎,𝑏)   𝑊(𝑞,𝑝,𝑎,𝑏)

Proof of Theorem imasaddflemg

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		imasaddfnlemg.v	. . 3 ⊢ (𝜑 → 𝑉 ∈ 𝑊)
5		imasaddfnlemg.x	. . 3 ⊢ (𝜑 → · ∈ 𝐶)
6	1, 2, 3, 4, 5	imasaddfnlemg 13544	. 2 ⊢ (𝜑 → ∙ Fn (𝐵 × 𝐵))
7		fof 5592	. . . . . . . . . 10 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹:𝑉⟶𝐵)
8	1, 7	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑉⟶𝐵)
9		ffvelcdm 5812	. . . . . . . . . . 11 ⊢ ((𝐹:𝑉⟶𝐵 ∧ 𝑝 ∈ 𝑉) → (𝐹‘𝑝) ∈ 𝐵)
10		ffvelcdm 5812	. . . . . . . . . . 11 ⊢ ((𝐹:𝑉⟶𝐵 ∧ 𝑞 ∈ 𝑉) → (𝐹‘𝑞) ∈ 𝐵)
11	9, 10	anim12dan 604	. . . . . . . . . 10 ⊢ ((𝐹:𝑉⟶𝐵 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → ((𝐹‘𝑝) ∈ 𝐵 ∧ (𝐹‘𝑞) ∈ 𝐵))
12		opelxpi 4783	. . . . . . . . . 10 ⊢ (((𝐹‘𝑝) ∈ 𝐵 ∧ (𝐹‘𝑞) ∈ 𝐵) → ⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩ ∈ (𝐵 × 𝐵))
13	11, 12	syl 14	. . . . . . . . 9 ⊢ ((𝐹:𝑉⟶𝐵 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → ⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩ ∈ (𝐵 × 𝐵))
14	8, 13	sylan 283	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → ⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩ ∈ (𝐵 × 𝐵))
15		imasaddflem.c	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)
16		ffvelcdm 5812	. . . . . . . . 9 ⊢ ((𝐹:𝑉⟶𝐵 ∧ (𝑝 · 𝑞) ∈ 𝑉) → (𝐹‘(𝑝 · 𝑞)) ∈ 𝐵)
17	8, 15, 16	syl2an2r 599	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝐹‘(𝑝 · 𝑞)) ∈ 𝐵)
18	14, 17	opelxpd 4784	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ ∈ ((𝐵 × 𝐵) × 𝐵))
19	18	snssd 3841	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × 𝐵))
20	19	anassrs 400	. . . . 5 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × 𝐵))
21	20	iunssd 4039	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑉) → ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × 𝐵))
22	21	iunssd 4039	. . 3 ⊢ (𝜑 → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} ⊆ ((𝐵 × 𝐵) × 𝐵))
23	3, 22	eqsstrd 3276	. 2 ⊢ (𝜑 → ∙ ⊆ ((𝐵 × 𝐵) × 𝐵))
24		dff2 5823	. 2 ⊢ ( ∙ :(𝐵 × 𝐵)⟶𝐵 ↔ ( ∙ Fn (𝐵 × 𝐵) ∧ ∙ ⊆ ((𝐵 × 𝐵) × 𝐵)))
25	6, 23, 24	sylanbrc 417	1 ⊢ (𝜑 → ∙ :(𝐵 × 𝐵)⟶𝐵)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398   ∈ wcel 2205   ⊆ wss 3213  {csn 3691  ⟨cop 3694  ∪ ciun 3993   × cxp 4749   Fn wfn 5349  ⟶wf 5350  –onto→wfo 5352  ‘cfv 5354  (class class class)co 6052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-ov 6055

This theorem is referenced by:  imasaddf  13549  imasmulf  13552  qusaddflemg  13564