Theorem mptmpoopabbrdOLD 8085

Description: Obsolete version of mptmpoopabbrd 8084 as of 7-Apr-2025. (Contributed by Alexander van Vekens, 8-Nov-2017.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
mptmpoopabbrd.g	⊢ (𝜑 → 𝐺 ∈ 𝑊)
mptmpoopabbrd.x	⊢ (𝜑 → 𝑋 ∈ (𝐴‘𝐺))
mptmpoopabbrd.y	⊢ (𝜑 → 𝑌 ∈ (𝐵‘𝐺))
mptmpoopabbrd.1	⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → (𝜏 ↔ 𝜃))
mptmpoopabbrd.2	⊢ (𝑔 = 𝐺 → (𝜒 ↔ 𝜏))
mptmpoopabbrd.m	⊢ 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴‘𝑔), 𝑏 ∈ (𝐵‘𝑔) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜒 ∧ 𝑓(𝐷‘𝑔)ℎ)}))

Assertion

Ref	Expression
mptmpoopabbrdOLD	⊢ (𝜑 → (𝑋(𝑀‘𝐺)𝑌) = {⟨𝑓, ℎ⟩ ∣ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)})

Distinct variable groups:   𝐴,𝑎,𝑏,𝑔   𝐵,𝑎,𝑏,𝑔   𝐷,𝑎,𝑏,𝑓,𝑔,ℎ   𝐺,𝑎,𝑏,𝑓,𝑔,ℎ   𝑔,𝑊   𝑋,𝑎,𝑏,𝑓,𝑔,ℎ   𝑌,𝑎,𝑏,𝑓,𝑔,ℎ   𝜑,𝑓,ℎ   𝜏,𝑔   𝜃,𝑎,𝑏

Allowed substitution hints:   𝜑(𝑔,𝑎,𝑏)   𝜒(𝑓,𝑔,ℎ,𝑎,𝑏)   𝜃(𝑓,𝑔,ℎ)   𝜏(𝑓,ℎ,𝑎,𝑏)   𝐴(𝑓,ℎ)   𝐵(𝑓,ℎ)   𝑀(𝑓,𝑔,ℎ,𝑎,𝑏)   𝑊(𝑓,ℎ,𝑎,𝑏)

Proof of Theorem mptmpoopabbrdOLD

Step	Hyp	Ref	Expression
1		mptmpoopabbrd.m	. . . 4 ⊢ 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴‘𝑔), 𝑏 ∈ (𝐵‘𝑔) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜒 ∧ 𝑓(𝐷‘𝑔)ℎ)}))
2		fveq2 6881	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝐴‘𝑔) = (𝐴‘𝐺))
3		fveq2 6881	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝐵‘𝑔) = (𝐵‘𝐺))
4		mptmpoopabbrd.2	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝜒 ↔ 𝜏))
5		fveq2 6881	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝐷‘𝑔) = (𝐷‘𝐺))
6	5	breqd 5135	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑓(𝐷‘𝑔)ℎ ↔ 𝑓(𝐷‘𝐺)ℎ))
7	4, 6	anbi12d 632	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝜒 ∧ 𝑓(𝐷‘𝑔)ℎ) ↔ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)))
8	7	opabbidv 5190	. . . . 5 ⊢ (𝑔 = 𝐺 → {⟨𝑓, ℎ⟩ ∣ (𝜒 ∧ 𝑓(𝐷‘𝑔)ℎ)} = {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)})
9	2, 3, 8	mpoeq123dv 7487	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑎 ∈ (𝐴‘𝑔), 𝑏 ∈ (𝐵‘𝑔) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜒 ∧ 𝑓(𝐷‘𝑔)ℎ)}) = (𝑎 ∈ (𝐴‘𝐺), 𝑏 ∈ (𝐵‘𝐺) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)}))
10		mptmpoopabbrd.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝑊)
11	10	elexd 3488	. . . 4 ⊢ (𝜑 → 𝐺 ∈ V)
12		fvex 6894	. . . . . 6 ⊢ (𝐴‘𝐺) ∈ V
13		fvex 6894	. . . . . 6 ⊢ (𝐵‘𝐺) ∈ V
14	12, 13	mpoex 8083	. . . . 5 ⊢ (𝑎 ∈ (𝐴‘𝐺), 𝑏 ∈ (𝐵‘𝐺) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)}) ∈ V
15	14	a1i 11	. . . 4 ⊢ (𝜑 → (𝑎 ∈ (𝐴‘𝐺), 𝑏 ∈ (𝐵‘𝐺) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)}) ∈ V)
16	1, 9, 11, 15	fvmptd3 7014	. . 3 ⊢ (𝜑 → (𝑀‘𝐺) = (𝑎 ∈ (𝐴‘𝐺), 𝑏 ∈ (𝐵‘𝐺) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)}))
17	16	oveqd 7427	. 2 ⊢ (𝜑 → (𝑋(𝑀‘𝐺)𝑌) = (𝑋(𝑎 ∈ (𝐴‘𝐺), 𝑏 ∈ (𝐵‘𝐺) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)})𝑌))
18		mptmpoopabbrd.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐴‘𝐺))
19		mptmpoopabbrd.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝐵‘𝐺))
20		mptmpoopabbrd.1	. . . . . 6 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → (𝜏 ↔ 𝜃))
21	20	anbi1d 631	. . . . 5 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → ((𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ) ↔ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)))
22	21	opabbidv 5190	. . . 4 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)} = {⟨𝑓, ℎ⟩ ∣ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)})
23		eqid 2736	. . . 4 ⊢ (𝑎 ∈ (𝐴‘𝐺), 𝑏 ∈ (𝐵‘𝐺) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)}) = (𝑎 ∈ (𝐴‘𝐺), 𝑏 ∈ (𝐵‘𝐺) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)})
24		ancom 460	. . . . . 6 ⊢ ((𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ) ↔ (𝑓(𝐷‘𝐺)ℎ ∧ 𝜃))
25	24	opabbii 5191	. . . . 5 ⊢ {⟨𝑓, ℎ⟩ ∣ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)} = {⟨𝑓, ℎ⟩ ∣ (𝑓(𝐷‘𝐺)ℎ ∧ 𝜃)}
26		opabresex2 7464	. . . . 5 ⊢ {⟨𝑓, ℎ⟩ ∣ (𝑓(𝐷‘𝐺)ℎ ∧ 𝜃)} ∈ V
27	25, 26	eqeltri 2831	. . . 4 ⊢ {⟨𝑓, ℎ⟩ ∣ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)} ∈ V
28	22, 23, 27	ovmpoa 7567	. . 3 ⊢ ((𝑋 ∈ (𝐴‘𝐺) ∧ 𝑌 ∈ (𝐵‘𝐺)) → (𝑋(𝑎 ∈ (𝐴‘𝐺), 𝑏 ∈ (𝐵‘𝐺) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)})𝑌) = {⟨𝑓, ℎ⟩ ∣ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)})
29	18, 19, 28	syl2anc 584	. 2 ⊢ (𝜑 → (𝑋(𝑎 ∈ (𝐴‘𝐺), 𝑏 ∈ (𝐵‘𝐺) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)})𝑌) = {⟨𝑓, ℎ⟩ ∣ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)})
30	17, 29	eqtrd 2771	1 ⊢ (𝜑 → (𝑋(𝑀‘𝐺)𝑌) = {⟨𝑓, ℎ⟩ ∣ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3464   class class class wbr 5124  {copab 5186   ↦ cmpt 5206  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994

This theorem is referenced by: (None)