Theorem imasaddvallemg 12742

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 5881	. 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 12741	. . . . 5 ⊢ (𝜑 → ∙ Fn (𝐵 × 𝐵))
8		fnfun 5315	. . . . 5 ⊢ ( ∙ Fn (𝐵 × 𝐵) → Fun ∙ )
9	7, 8	syl 14	. . . 4 ⊢ (𝜑 → Fun ∙ )
10	9	3ad2ant1 1018	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → Fun ∙ )
11		fveq2 5517	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑋 → (𝐹‘𝑝) = (𝐹‘𝑋))
12	11	opeq1d 3786	. . . . . . . . . 10 ⊢ (𝑝 = 𝑋 → ⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩ = ⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩)
13		fvoveq1 5901	. . . . . . . . . 10 ⊢ (𝑝 = 𝑋 → (𝐹‘(𝑝 · 𝑌)) = (𝐹‘(𝑋 · 𝑌)))
14	12, 13	opeq12d 3788	. . . . . . . . 9 ⊢ (𝑝 = 𝑋 → ⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩ = ⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩)
15	14	sneqd 3607	. . . . . . . 8 ⊢ (𝑝 = 𝑋 → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} = {⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩})
16	15	ssiun2s 3932	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → {⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ ∪ 𝑝 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩})
17	16	3ad2ant2 1019	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ ∪ 𝑝 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩})
18		fveq2 5517	. . . . . . . . . . . . 13 ⊢ (𝑞 = 𝑌 → (𝐹‘𝑞) = (𝐹‘𝑌))
19	18	opeq2d 3787	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑌 → ⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩ = ⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩)
20		oveq2 5886	. . . . . . . . . . . . 13 ⊢ (𝑞 = 𝑌 → (𝑝 · 𝑞) = (𝑝 · 𝑌))
21	20	fveq2d 5521	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑌 → (𝐹‘(𝑝 · 𝑞)) = (𝐹‘(𝑝 · 𝑌)))
22	19, 21	opeq12d 3788	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑌 → ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ = ⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩)
23	22	sneqd 3607	. . . . . . . . . 10 ⊢ (𝑞 = 𝑌 → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} = {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩})
24	23	ssiun2s 3932	. . . . . . . . 9 ⊢ (𝑌 ∈ 𝑉 → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
25	24	ralrimivw 2551	. . . . . . . 8 ⊢ (𝑌 ∈ 𝑉 → ∀𝑝 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
26		ss2iun 3903	. . . . . . . 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 1020	. . . . . 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 3167	. . . . 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 1018	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
31	29, 30	sseqtrrd 3196	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ ∙ )
32		fof 5440	. . . . . . . . . . 11 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹:𝑉⟶𝐵)
33	2, 32	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝑉⟶𝐵)
34	33	3ad2ant1 1018	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝐹:𝑉⟶𝐵)
35	5	3ad2ant1 1018	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝑉 ∈ 𝑊)
36	34, 35	fexd 5749	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝐹 ∈ V)
37		simp2 998	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝑋 ∈ 𝑉)
38		fvexg 5536	. . . . . . . 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 999	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝑌 ∈ 𝑉)
41		fvexg 5536	. . . . . . . 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 4230	. . . . . . 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 1018	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → · ∈ 𝐶)
46		ovexg 5912	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ · ∈ 𝐶 ∧ 𝑌 ∈ 𝑉) → (𝑋 · 𝑌) ∈ V)
47	37, 45, 40, 46	syl3anc 1238	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 · 𝑌) ∈ V)
48		fvexg 5536	. . . . . . 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 4230	. . . . . 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 3728	. . . . 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 5558	. . 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 2222	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝐹‘𝑋) ∙ (𝐹‘𝑌)) = (𝐹‘(𝑋 · 𝑌)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  ∀wral 2455  Vcvv 2739   ⊆ wss 3131  {csn 3594  ⟨cop 3597  ∪ ciun 3888   × cxp 4626  Fun wfun 5212   Fn wfn 5213  ⟶wf 5214  –onto→wfo 5216  ‘cfv 5218  (class class class)co 5878

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5881

This theorem is referenced by:  imasaddval  12745  imasmulval  12748  qusaddvallemg  12758