Theorem imasaddvallemg 13461

Description: The operation of an image structure is defined to distribute over the mapping function. (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)

Assertion

Ref	Expression
imasaddvallemg	⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝐹‘𝑋) ∙ (𝐹‘𝑌)) = (𝐹‘(𝑋 · 𝑌)))

Distinct variable groups:   𝑞,𝑝,𝐵   𝑎,𝑏,𝑝,𝑞,𝑉   · ,𝑝,𝑞   𝑋,𝑝   𝐹,𝑎,𝑏,𝑝,𝑞   𝜑,𝑎,𝑏,𝑝,𝑞   ∙ ,𝑎,𝑏,𝑝,𝑞   𝑌,𝑝,𝑞

Allowed substitution hints:   𝐵(𝑎,𝑏)   𝐶(𝑞,𝑝,𝑎,𝑏)   · (𝑎,𝑏)   𝑊(𝑞,𝑝,𝑎,𝑏)   𝑋(𝑞,𝑎,𝑏)   𝑌(𝑎,𝑏)

Proof of Theorem imasaddvallemg

Step	Hyp	Ref	Expression
1		df-ov 6031	. 2 ⊢ ((𝐹‘𝑋) ∙ (𝐹‘𝑌)) = ( ∙ ‘⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩)
2		imasaddf.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
3		imasaddf.e	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))
4		imasaddflem.a	. . . . . 6 ⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
5		imasaddfnlemg.v	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ 𝑊)
6		imasaddfnlemg.x	. . . . . 6 ⊢ (𝜑 → · ∈ 𝐶)
7	2, 3, 4, 5, 6	imasaddfnlemg 13460	. . . . 5 ⊢ (𝜑 → ∙ Fn (𝐵 × 𝐵))
8		fnfun 5434	. . . . 5 ⊢ ( ∙ Fn (𝐵 × 𝐵) → Fun ∙ )
9	7, 8	syl 14	. . . 4 ⊢ (𝜑 → Fun ∙ )
10	9	3ad2ant1 1045	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → Fun ∙ )
11		fveq2 5648	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑋 → (𝐹‘𝑝) = (𝐹‘𝑋))
12	11	opeq1d 3873	. . . . . . . . . 10 ⊢ (𝑝 = 𝑋 → ⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩ = ⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩)
13		fvoveq1 6051	. . . . . . . . . 10 ⊢ (𝑝 = 𝑋 → (𝐹‘(𝑝 · 𝑌)) = (𝐹‘(𝑋 · 𝑌)))
14	12, 13	opeq12d 3875	. . . . . . . . 9 ⊢ (𝑝 = 𝑋 → ⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩ = ⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩)
15	14	sneqd 3686	. . . . . . . 8 ⊢ (𝑝 = 𝑋 → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} = {⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩})
16	15	ssiun2s 4019	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → {⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ ∪ 𝑝 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩})
17	16	3ad2ant2 1046	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ ∪ 𝑝 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩})
18		fveq2 5648	. . . . . . . . . . . . 13 ⊢ (𝑞 = 𝑌 → (𝐹‘𝑞) = (𝐹‘𝑌))
19	18	opeq2d 3874	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑌 → ⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩ = ⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩)
20		oveq2 6036	. . . . . . . . . . . . 13 ⊢ (𝑞 = 𝑌 → (𝑝 · 𝑞) = (𝑝 · 𝑌))
21	20	fveq2d 5652	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑌 → (𝐹‘(𝑝 · 𝑞)) = (𝐹‘(𝑝 · 𝑌)))
22	19, 21	opeq12d 3875	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑌 → ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ = ⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩)
23	22	sneqd 3686	. . . . . . . . . 10 ⊢ (𝑞 = 𝑌 → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} = {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩})
24	23	ssiun2s 4019	. . . . . . . . 9 ⊢ (𝑌 ∈ 𝑉 → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
25	24	ralrimivw 2607	. . . . . . . 8 ⊢ (𝑌 ∈ 𝑉 → ∀𝑝 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
26		ss2iun 3990	. . . . . . . 8 ⊢ (∀𝑝 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} → ∪ 𝑝 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
27	25, 26	syl 14	. . . . . . 7 ⊢ (𝑌 ∈ 𝑉 → ∪ 𝑝 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
28	27	3ad2ant3 1047	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ∪ 𝑝 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
29	17, 28	sstrd 3238	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
30	4	3ad2ant1 1045	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
31	29, 30	sseqtrrd 3267	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ ∙ )
32		fof 5568	. . . . . . . . . . 11 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹:𝑉⟶𝐵)
33	2, 32	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝑉⟶𝐵)
34	33	3ad2ant1 1045	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝐹:𝑉⟶𝐵)
35	5	3ad2ant1 1045	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝑉 ∈ 𝑊)
36	34, 35	fexd 5894	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝐹 ∈ V)
37		simp2 1025	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝑋 ∈ 𝑉)
38		fvexg 5667	. . . . . . . 8 ⊢ ((𝐹 ∈ V ∧ 𝑋 ∈ 𝑉) → (𝐹‘𝑋) ∈ V)
39	36, 37, 38	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝐹‘𝑋) ∈ V)
40		simp3 1026	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝑌 ∈ 𝑉)
41		fvexg 5667	. . . . . . . 8 ⊢ ((𝐹 ∈ V ∧ 𝑌 ∈ 𝑉) → (𝐹‘𝑌) ∈ V)
42	36, 40, 41	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝐹‘𝑌) ∈ V)
43		opexg 4326	. . . . . . 7 ⊢ (((𝐹‘𝑋) ∈ V ∧ (𝐹‘𝑌) ∈ V) → ⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩ ∈ V)
44	39, 42, 43	syl2anc 411	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩ ∈ V)
45	6	3ad2ant1 1045	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → · ∈ 𝐶)
46		ovexg 6062	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ · ∈ 𝐶 ∧ 𝑌 ∈ 𝑉) → (𝑋 · 𝑌) ∈ V)
47	37, 45, 40, 46	syl3anc 1274	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 · 𝑌) ∈ V)
48		fvexg 5667	. . . . . . 7 ⊢ ((𝐹 ∈ V ∧ (𝑋 · 𝑌) ∈ V) → (𝐹‘(𝑋 · 𝑌)) ∈ V)
49	36, 47, 48	syl2anc 411	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝐹‘(𝑋 · 𝑌)) ∈ V)
50		opexg 4326	. . . . . 6 ⊢ ((⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩ ∈ V ∧ (𝐹‘(𝑋 · 𝑌)) ∈ V) → ⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ V)
51	44, 49, 50	syl2anc 411	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ V)
52		snssg 3812	. . . . 5 ⊢ (⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ V → (⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ ∙ ↔ {⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ ∙ ))
53	51, 52	syl 14	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ ∙ ↔ {⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ ∙ ))
54	31, 53	mpbird 167	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ ∙ )
55		funopfv 5692	. . 3 ⊢ (Fun ∙ → (⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ ∙ → ( ∙ ‘⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩) = (𝐹‘(𝑋 · 𝑌))))
56	10, 54, 55	sylc 62	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ( ∙ ‘⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩) = (𝐹‘(𝑋 · 𝑌)))
57	1, 56	eqtrid 2276	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝐹‘𝑋) ∙ (𝐹‘𝑌)) = (𝐹‘(𝑋 · 𝑌)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398   ∈ wcel 2202  ∀wral 2511  Vcvv 2803   ⊆ wss 3201  {csn 3673  ⟨cop 3676  ∪ ciun 3975   × cxp 4729  Fun wfun 5327   Fn wfn 5328  ⟶wf 5329  –onto→wfo 5331  ‘cfv 5333  (class class class)co 6028

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031

This theorem is referenced by:  imasaddval  13464  imasmulval  13467  qusaddvallemg  13479