Theorem mptmpoopabbrdOLDOLD 8124

Description: Obsolete version of mptmpoopabbrd 8121 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
mptmpoopabbrdOLDOLD	⊢ (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 mptmpoopabbrdOLDOLD

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 6920	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝐴‘𝑔) = (𝐴‘𝐺))
4		fveq2 6920	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝐵‘𝑔) = (𝐵‘𝐺))
5		mptmpoopabbrdOLD.2	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝜒 ↔ 𝜏))
6		fveq2 6920	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝐷‘𝑔) = (𝐷‘𝐺))
7	6	breqd 5177	. . . . . . . 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 5232	. . . . . 6 ⊢ (𝑔 = 𝐺 → {⟨𝑓, ℎ⟩ ∣ (𝜒 ∧ 𝑓(𝐷‘𝑔)ℎ)} = {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)})
10	3, 4, 9	mpoeq123dv 7525	. . . . 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 3509	. . . . . 6 ⊢ (𝐺 ∈ 𝑊 → 𝐺 ∈ V)
12	11	adantr 480	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐺 ∈ 𝑊) → 𝐺 ∈ V)
13		fvex 6933	. . . . . . 7 ⊢ (𝐴‘𝐺) ∈ V
14		fvex 6933	. . . . . . 7 ⊢ (𝐵‘𝐺) ∈ V
15	13, 14	pm3.2i 470	. . . . . 6 ⊢ ((𝐴‘𝐺) ∈ V ∧ (𝐵‘𝐺) ∈ V)
16		mpoexga 8118	. . . . . 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 7052	. . . 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 583	. . 3 ⊢ (𝜑 → (𝑀‘𝐺) = (𝑎 ∈ (𝐴‘𝐺), 𝑏 ∈ (𝐵‘𝐺) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)}))
20	19	oveqd 7465	. 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 460	. . . . 5 ⊢ ((𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ) ↔ (𝑓(𝐷‘𝐺)ℎ ∧ 𝜃))
24	23	opabbii 5233	. . . 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 7503	. . . 4 ⊢ (𝜑 → {⟨𝑓, ℎ⟩ ∣ (𝑓(𝐷‘𝐺)ℎ ∧ 𝜃)} ∈ V)
28	24, 27	eqeltrid 2848	. . 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 5232	. . . 4 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)} = {⟨𝑓, ℎ⟩ ∣ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)})
32		eqid 2740	. . . 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 7604	. . 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 1371	. 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 2780	1 ⊢ (𝜑 → (𝑋(𝑀‘𝐺)𝑌) = {⟨𝑓, ℎ⟩ ∣ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488   class class class wbr 5166  {copab 5228   ↦ cmpt 5249  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450

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-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

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-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  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-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031

This theorem is referenced by:  mptmpoopabovdOLD  8125