Theorem imasaddvallem 17589

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 7451	. 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 17588	. . . . 5 ⊢ (𝜑 → ∙ Fn (𝐵 × 𝐵))
6		fnfun 6679	. . . . 5 ⊢ ( ∙ Fn (𝐵 × 𝐵) → Fun ∙ )
7	5, 6	syl 17	. . . 4 ⊢ (𝜑 → Fun ∙ )
8	7	3ad2ant1 1133	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → Fun ∙ )
9		fveq2 6920	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑋 → (𝐹‘𝑝) = (𝐹‘𝑋))
10	9	opeq1d 4903	. . . . . . . . . 10 ⊢ (𝑝 = 𝑋 → ⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩ = ⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩)
11		fvoveq1 7471	. . . . . . . . . 10 ⊢ (𝑝 = 𝑋 → (𝐹‘(𝑝 · 𝑌)) = (𝐹‘(𝑋 · 𝑌)))
12	10, 11	opeq12d 4905	. . . . . . . . 9 ⊢ (𝑝 = 𝑋 → ⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩ = ⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩)
13	12	sneqd 4660	. . . . . . . 8 ⊢ (𝑝 = 𝑋 → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} = {⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩})
14	13	ssiun2s 5071	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → {⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ ∪ 𝑝 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩})
15	14	3ad2ant2 1134	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ ∪ 𝑝 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩})
16		fveq2 6920	. . . . . . . . . . . . 13 ⊢ (𝑞 = 𝑌 → (𝐹‘𝑞) = (𝐹‘𝑌))
17	16	opeq2d 4904	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑌 → ⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩ = ⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩)
18		oveq2 7456	. . . . . . . . . . . . 13 ⊢ (𝑞 = 𝑌 → (𝑝 · 𝑞) = (𝑝 · 𝑌))
19	18	fveq2d 6924	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑌 → (𝐹‘(𝑝 · 𝑞)) = (𝐹‘(𝑝 · 𝑌)))
20	17, 19	opeq12d 4905	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑌 → ⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩ = ⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩)
21	20	sneqd 4660	. . . . . . . . . 10 ⊢ (𝑞 = 𝑌 → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩} = {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩})
22	21	ssiun2s 5071	. . . . . . . . 9 ⊢ (𝑌 ∈ 𝑉 → {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
23	22	ralrimivw 3156	. . . . . . . 8 ⊢ (𝑌 ∈ 𝑉 → ∀𝑝 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑌)⟩, (𝐹‘(𝑝 · 𝑌))⟩} ⊆ ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
24		ss2iun 5033	. . . . . . . 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 1135	. . . . . 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 4019	. . . . 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 1133	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {⟨⟨(𝐹‘𝑝), (𝐹‘𝑞)⟩, (𝐹‘(𝑝 · 𝑞))⟩})
29	27, 28	sseqtrrd 4050	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ ∙ )
30		opex 5484	. . . . 5 ⊢ ⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ V
31	30	snss 4810	. . . 4 ⊢ (⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ ∙ ↔ {⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩} ⊆ ∙ )
32	29, 31	sylibr 234	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ⟨⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩, (𝐹‘(𝑋 · 𝑌))⟩ ∈ ∙ )
33		funopfv 6972	. . 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 2792	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝐹‘𝑋) ∙ (𝐹‘𝑌)) = (𝐹‘(𝑋 · 𝑌)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ⊆ wss 3976  {csn 4648  ⟨cop 4654  ∪ ciun 5015   × cxp 5698  Fun wfun 6567   Fn wfn 6568  –onto→wfo 6571  ‘cfv 6573  (class class class)co 7448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fo 6579  df-fv 6581  df-ov 7451

This theorem is referenced by:  imasaddval  17592  imasmulval  17595  qusaddvallem  17611