Theorem mptmpoopabbrd 8105

Description: The operation value of a function value of a collection of ordered pairs of elements related in two ways. (Contributed by Alexander van Vekens, 8-Nov-2017.) (Revised by AV, 15-Jan-2021.) Add disjoint variable condition on 𝐷, 𝑓, ℎ to remove hypotheses; avoid ax-rep 5279. (Revised by SN, 7-Apr-2025.)

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
mptmpoopabbrd	⊢ (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 mptmpoopabbrd

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 6906	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝐴‘𝑔) = (𝐴‘𝐺))
3		fveq2 6906	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝐵‘𝑔) = (𝐵‘𝐺))
4		mptmpoopabbrd.2	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝜒 ↔ 𝜏))
5		fveq2 6906	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝐷‘𝑔) = (𝐷‘𝐺))
6	5	breqd 5154	. . . . . . 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 5209	. . . . 5 ⊢ (𝑔 = 𝐺 → {⟨𝑓, ℎ⟩ ∣ (𝜒 ∧ 𝑓(𝐷‘𝑔)ℎ)} = {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)})
9	2, 3, 8	mpoeq123dv 7508	. . . 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 3504	. . . 4 ⊢ (𝜑 → 𝐺 ∈ V)
12		fvex 6919	. . . . . 6 ⊢ (𝐴‘𝐺) ∈ V
13		fvex 6919	. . . . . 6 ⊢ (𝐵‘𝐺) ∈ V
14		fvex 6919	. . . . . . 7 ⊢ (𝐷‘𝐺) ∈ V
15	14	pwex 5380	. . . . . 6 ⊢ 𝒫 (𝐷‘𝐺) ∈ V
16		simpr 484	. . . . . . . . . 10 ⊢ ((𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ) → 𝑓(𝐷‘𝐺)ℎ)
17	16	ssopab2i 5555	. . . . . . . . 9 ⊢ {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)} ⊆ {⟨𝑓, ℎ⟩ ∣ 𝑓(𝐷‘𝐺)ℎ}
18		opabss 5207	. . . . . . . . 9 ⊢ {⟨𝑓, ℎ⟩ ∣ 𝑓(𝐷‘𝐺)ℎ} ⊆ (𝐷‘𝐺)
19	17, 18	sstri 3993	. . . . . . . 8 ⊢ {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)} ⊆ (𝐷‘𝐺)
20	14, 19	elpwi2 5335	. . . . . . 7 ⊢ {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)} ∈ 𝒫 (𝐷‘𝐺)
21	20	rgen2w 3066	. . . . . 6 ⊢ ∀𝑎 ∈ (𝐴‘𝐺)∀𝑏 ∈ (𝐵‘𝐺){⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)} ∈ 𝒫 (𝐷‘𝐺)
22	12, 13, 15, 21	mpoexw 8103	. . . . 5 ⊢ (𝑎 ∈ (𝐴‘𝐺), 𝑏 ∈ (𝐵‘𝐺) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)}) ∈ V
23	22	a1i 11	. . . 4 ⊢ (𝜑 → (𝑎 ∈ (𝐴‘𝐺), 𝑏 ∈ (𝐵‘𝐺) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)}) ∈ V)
24	1, 9, 11, 23	fvmptd3 7039	. . 3 ⊢ (𝜑 → (𝑀‘𝐺) = (𝑎 ∈ (𝐴‘𝐺), 𝑏 ∈ (𝐵‘𝐺) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)}))
25	24	oveqd 7448	. 2 ⊢ (𝜑 → (𝑋(𝑀‘𝐺)𝑌) = (𝑋(𝑎 ∈ (𝐴‘𝐺), 𝑏 ∈ (𝐵‘𝐺) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)})𝑌))
26		mptmpoopabbrd.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐴‘𝐺))
27		mptmpoopabbrd.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝐵‘𝐺))
28		mptmpoopabbrd.1	. . . . . 6 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → (𝜏 ↔ 𝜃))
29	28	anbi1d 631	. . . . 5 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → ((𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ) ↔ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)))
30	29	opabbidv 5209	. . . 4 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)} = {⟨𝑓, ℎ⟩ ∣ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)})
31		eqid 2737	. . . 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)})
32		ancom 460	. . . . . 6 ⊢ ((𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ) ↔ (𝑓(𝐷‘𝐺)ℎ ∧ 𝜃))
33	32	opabbii 5210	. . . . 5 ⊢ {⟨𝑓, ℎ⟩ ∣ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)} = {⟨𝑓, ℎ⟩ ∣ (𝑓(𝐷‘𝐺)ℎ ∧ 𝜃)}
34		opabresex2 7485	. . . . 5 ⊢ {⟨𝑓, ℎ⟩ ∣ (𝑓(𝐷‘𝐺)ℎ ∧ 𝜃)} ∈ V
35	33, 34	eqeltri 2837	. . . 4 ⊢ {⟨𝑓, ℎ⟩ ∣ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)} ∈ V
36	30, 31, 35	ovmpoa 7588	. . 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)})
37	26, 27, 36	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)})
38	25, 37	eqtrd 2777	1 ⊢ (𝜑 → (𝑋(𝑀‘𝐺)𝑌) = {⟨𝑓, ℎ⟩ ∣ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3480  𝒫 cpw 4600   class class class wbr 5143  {copab 5205   ↦ cmpt 5225  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015

This theorem is referenced by:  mptmpoopabovd  8107  wlkson  29674