Theorem mptmpoopabbrdOLD 7923

Description: Obsolete version of mptmpoopabbrd 7921 as of 13-Dec-2024. (Contributed by Alexander van Vekens, 8-Nov-2017.) (Revised by AV, 15-Jan-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Proof of Theorem mptmpoopabbrdOLD

Step	Hyp	Ref	Expression
1		mptmpoopabbrdOLD.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑊)
2		mptmpoopabbrdOLD.m	. . . . 5 ⊢ 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴‘𝑔), 𝑏 ∈ (𝐵‘𝑔) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜒 ∧ 𝑓(𝐷‘𝑔)ℎ)}))
3		fveq2 6774	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝐴‘𝑔) = (𝐴‘𝐺))
4		fveq2 6774	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝐵‘𝑔) = (𝐵‘𝐺))
5		mptmpoopabbrdOLD.2	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝜒 ↔ 𝜏))
6		fveq2 6774	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝐷‘𝑔) = (𝐷‘𝐺))
7	6	breqd 5085	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑓(𝐷‘𝑔)ℎ ↔ 𝑓(𝐷‘𝐺)ℎ))
8	5, 7	anbi12d 631	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((𝜒 ∧ 𝑓(𝐷‘𝑔)ℎ) ↔ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)))
9	8	opabbidv 5140	. . . . . 6 ⊢ (𝑔 = 𝐺 → {⟨𝑓, ℎ⟩ ∣ (𝜒 ∧ 𝑓(𝐷‘𝑔)ℎ)} = {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)})
10	3, 4, 9	mpoeq123dv 7350	. . . . 5 ⊢ (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)}))
11		elex 3450	. . . . . 6 ⊢ (𝐺 ∈ 𝑊 → 𝐺 ∈ V)
12	11	adantr 481	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐺 ∈ 𝑊) → 𝐺 ∈ V)
13		fvex 6787	. . . . . . 7 ⊢ (𝐴‘𝐺) ∈ V
14		fvex 6787	. . . . . . 7 ⊢ (𝐵‘𝐺) ∈ V
15	13, 14	pm3.2i 471	. . . . . 6 ⊢ ((𝐴‘𝐺) ∈ V ∧ (𝐵‘𝐺) ∈ V)
16		mpoexga 7918	. . . . . 6 ⊢ (((𝐴‘𝐺) ∈ V ∧ (𝐵‘𝐺) ∈ V) → (𝑎 ∈ (𝐴‘𝐺), 𝑏 ∈ (𝐵‘𝐺) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)}) ∈ V)
17	15, 16	mp1i 13	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐺 ∈ 𝑊) → (𝑎 ∈ (𝐴‘𝐺), 𝑏 ∈ (𝐵‘𝐺) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)}) ∈ V)
18	2, 10, 12, 17	fvmptd3 6898	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐺 ∈ 𝑊) → (𝑀‘𝐺) = (𝑎 ∈ (𝐴‘𝐺), 𝑏 ∈ (𝐵‘𝐺) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)}))
19	1, 1, 18	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑀‘𝐺) = (𝑎 ∈ (𝐴‘𝐺), 𝑏 ∈ (𝐵‘𝐺) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)}))
20	19	oveqd 7292	. 2 ⊢ (𝜑 → (𝑋(𝑀‘𝐺)𝑌) = (𝑋(𝑎 ∈ (𝐴‘𝐺), 𝑏 ∈ (𝐵‘𝐺) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)})𝑌))
21		mptmpoopabbrdOLD.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐴‘𝐺))
22		mptmpoopabbrdOLD.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝐵‘𝐺))
23		ancom 461	. . . . 5 ⊢ ((𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ) ↔ (𝑓(𝐷‘𝐺)ℎ ∧ 𝜃))
24	23	opabbii 5141	. . . 4 ⊢ {⟨𝑓, ℎ⟩ ∣ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)} = {⟨𝑓, ℎ⟩ ∣ (𝑓(𝐷‘𝐺)ℎ ∧ 𝜃)}
25		mptmpoopabbrdOLD.r	. . . . 5 ⊢ ((𝜑 ∧ 𝑓(𝐷‘𝐺)ℎ) → 𝜓)
26		mptmpoopabbrdOLD.v	. . . . 5 ⊢ (𝜑 → {⟨𝑓, ℎ⟩ ∣ 𝜓} ∈ 𝑉)
27	25, 26	opabresex2d 7328	. . . 4 ⊢ (𝜑 → {⟨𝑓, ℎ⟩ ∣ (𝑓(𝐷‘𝐺)ℎ ∧ 𝜃)} ∈ V)
28	24, 27	eqeltrid 2843	. . 3 ⊢ (𝜑 → {⟨𝑓, ℎ⟩ ∣ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)} ∈ V)
29		mptmpoopabbrdOLD.1	. . . . . 6 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → (𝜏 ↔ 𝜃))
30	29	anbi1d 630	. . . . 5 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → ((𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ) ↔ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)))
31	30	opabbidv 5140	. . . 4 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)} = {⟨𝑓, ℎ⟩ ∣ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)})
32		eqid 2738	. . . 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)})
33	31, 32	ovmpoga 7427	. . 3 ⊢ ((𝑋 ∈ (𝐴‘𝐺) ∧ 𝑌 ∈ (𝐵‘𝐺) ∧ {⟨𝑓, ℎ⟩ ∣ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)} ∈ V) → (𝑋(𝑎 ∈ (𝐴‘𝐺), 𝑏 ∈ (𝐵‘𝐺) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)})𝑌) = {⟨𝑓, ℎ⟩ ∣ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)})
34	21, 22, 28, 33	syl3anc 1370	. 2 ⊢ (𝜑 → (𝑋(𝑎 ∈ (𝐴‘𝐺), 𝑏 ∈ (𝐵‘𝐺) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)})𝑌) = {⟨𝑓, ℎ⟩ ∣ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)})
35	20, 34	eqtrd 2778	1 ⊢ (𝜑 → (𝑋(𝑀‘𝐺)𝑌) = {⟨𝑓, ℎ⟩ ∣ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432   class class class wbr 5074  {copab 5136   ↦ cmpt 5157  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832

This theorem is referenced by:  mptmpoopabovdOLD  7924