Theorem imasaddvallem 17157

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 7258	. 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 17156	. . . . 5 ⊢ (𝜑 → ∙ Fn (𝐵 × 𝐵))
6		fnfun 6517	. . . . 5 ⊢ ( ∙ Fn (𝐵 × 𝐵) → Fun ∙ )
7	5, 6	syl 17	. . . 4 ⊢ (𝜑 → Fun ∙ )
8	7	3ad2ant1 1131	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → Fun ∙ )
9		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑋 → (𝐹‘𝑝) = (𝐹‘𝑋))
10	9	opeq1d 4807	. . . . . . . . . 10 ⊢ (𝑝 = 𝑋 → ⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩ = ⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩)
11		fvoveq1 7278	. . . . . . . . . 10 ⊢ (𝑝 = 𝑋 → (𝐹‘(𝑝 · 𝑌)) = (𝐹‘(𝑋 · 𝑌)))
12	10, 11	opeq12d 4809	. . . . . . . . 9 ⊢ (𝑝 = 𝑋 → ⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩ = ⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩)
13	12	sneqd 4570	. . . . . . . 8 ⊢ (𝑝 = 𝑋 → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} = {⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩})
14	13	ssiun2s 4974	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → {⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ ∪ 𝑝 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩})
15	14	3ad2ant2 1132	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ ∪ 𝑝 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩})
16		fveq2 6756	. . . . . . . . . . . . 13 ⊢ (𝑞 = 𝑌 → (𝐹‘𝑞) = (𝐹‘𝑌))
17	16	opeq2d 4808	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑌 → ⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩ = ⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩)
18		oveq2 7263	. . . . . . . . . . . . 13 ⊢ (𝑞 = 𝑌 → (𝑝 · 𝑞) = (𝑝 · 𝑌))
19	18	fveq2d 6760	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑌 → (𝐹‘(𝑝 · 𝑞)) = (𝐹‘(𝑝 · 𝑌)))
20	17, 19	opeq12d 4809	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑌 → ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ = ⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩)
21	20	sneqd 4570	. . . . . . . . . 10 ⊢ (𝑞 = 𝑌 → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} = {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩})
22	21	ssiun2s 4974	. . . . . . . . 9 ⊢ (𝑌 ∈ 𝑉 → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
23	22	ralrimivw 3108	. . . . . . . 8 ⊢ (𝑌 ∈ 𝑉 → ∀𝑝 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
24		ss2iun 4939	. . . . . . . 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 17	. . . . . . 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 1133	. . . . . 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 3927	. . . . 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 1131	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
29	27, 28	sseqtrrd 3958	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ ∙ )
30		opex 5373	. . . . 5 ⊢ ⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ V
31	30	snss 4716	. . . 4 ⊢ (⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ ∙ ↔ {⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ ∙ )
32	29, 31	sylibr 233	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ ∙ )
33		funopfv 6803	. . 3 ⊢ (Fun ∙ → (⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ ∙ → ( ∙ ‘⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩) = (𝐹‘(𝑋 · 𝑌))))
34	8, 32, 33	sylc 65	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ( ∙ ‘⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩) = (𝐹‘(𝑋 · 𝑌)))
35	1, 34	eqtrid 2790	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝐹‘𝑋) ∙ (𝐹‘𝑌)) = (𝐹‘(𝑋 · 𝑌)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063   ⊆ wss 3883  {csn 4558  ⟨cop 4564  ∪ ciun 4921   × cxp 5578  Fun wfun 6412   Fn wfn 6413  –onto→wfo 6416  ‘cfv 6418  (class class class)co 7255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fo 6424  df-fv 6426  df-ov 7258

This theorem is referenced by:  imasaddval  17160  imasmulval  17163  qusaddvallem  17179