Theorem imasaddvallem 17605

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))⟩})

Assertion

Ref	Expression
imasaddvallem	⊢ ((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)   · (𝑎, 𝑏)   𝑋(𝑞, 𝑎, 𝑏)   𝑌(𝑎, 𝑏)

Proof of Theorem imasaddvallem

Step	Hyp	Ref	Expression
1		df-ov 7422	. 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	2, 3, 4	imasaddfnlem 17604	. . . . 5 ⊢ (𝜑 → ∙ Fn (𝐵 × 𝐵))
6		fnfun 6639	. . . . 5 ⊢ ( ∙ Fn (𝐵 × 𝐵) → Fun ∙ )
7	5, 6	syl 18	. . . 4 ⊢ (𝜑 → Fun ∙ )
8	7	3ad2ant1 1151	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → Fun ∙ )
9		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑋 → (𝐹‘𝑝) = (𝐹‘𝑋))
10	9	opeq1d 4846	. . . . . . . . . 10 ⊢ (𝑝 = 𝑋 → ⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩ = ⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩)
11		fvoveq1 7442	. . . . . . . . . 10 ⊢ (𝑝 = 𝑋 → (𝐹‘(𝑝 · 𝑌)) = (𝐹‘(𝑋 · 𝑌)))
12	10, 11	opeq12d 4848	. . . . . . . . 9 ⊢ (𝑝 = 𝑋 → ⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩ = ⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩)
13	12	sneqd 4603	. . . . . . . 8 ⊢ (𝑝 = 𝑋 → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} = {⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩})
14	13	ssiun2s 5015	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → {⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ ∪ 𝑝 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩})
15	14	3ad2ant2 1152	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ ∪ 𝑝 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩})
16		fveq2 6885	. . . . . . . . . . . . 13 ⊢ (𝑞 = 𝑌 → (𝐹‘𝑞) = (𝐹‘𝑌))
17	16	opeq2d 4847	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑌 → ⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩ = ⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩)
18		oveq2 7427	. . . . . . . . . . . . 13 ⊢ (𝑞 = 𝑌 → (𝑝 · 𝑞) = (𝑝 · 𝑌))
19	18	fveq2d 6889	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑌 → (𝐹‘(𝑝 · 𝑞)) = (𝐹‘(𝑝 · 𝑌)))
20	17, 19	opeq12d 4848	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑌 → ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ = ⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩)
21	20	sneqd 4603	. . . . . . . . . 10 ⊢ (𝑞 = 𝑌 → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} = {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩})
22	21	ssiun2s 5015	. . . . . . . . 9 ⊢ (𝑌 ∈ 𝑉 → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
23	22	ralrimivw 3163	. . . . . . . 8 ⊢ (𝑌 ∈ 𝑉 → ∀𝑝 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
24		ss2iun 4977	. . . . . . . 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))⟩})
25	23, 24	syl 18	. . . . . . 7 ⊢ (𝑌 ∈ 𝑉 → ∪ 𝑝 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
26	25	3ad2ant3 1153	. . . . . 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))⟩})
27	15, 26	sstrd 3948	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
28	4	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
29	27, 28	sseqtrrd 3975	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ ∙ )
30		opex 5447	. . . . 5 ⊢ ⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ V
31	30	snss 4752	. . . 4 ⊢ (⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ ∙ ↔ {⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ ∙ )
32	29, 31	sylibr 237	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ ∙ )
33		funopfv 6934	. . 3 ⊢ (Fun ∙ → (⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ ∙ → ( ∙ ‘⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩) = (𝐹‘(𝑋 · 𝑌))))
34	8, 32, 33	sylc 66	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ( ∙ ‘⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩) = (𝐹‘(𝑋 · 𝑌)))
35	1, 34	eqtrid 2812	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝐹‘𝑋) ∙ (𝐹‘𝑌)) = (𝐹‘(𝑋 · 𝑌)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081   ⊆ wss 3906  {csn 4591  ⟨cop 4597  ∪ ciun 4958   × cxp 5661  Fun wfun 6534   Fn wfn 6535  –onto→wfo 6538  ‘cfv 6540  (class class class)co 7419

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fo 6546  df-fv 6548  df-ov 7422

This theorem is used by:  imasaddval  17608  imasmulval  17611  qusaddvallem  17627