Theorem mptmpoopabbrd 7797

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.)

Hypotheses

Ref	Expression
mptmpoopabbrd.g	⊢ (𝜑 → 𝐺 ∈ 𝑊)
mptmpoopabbrd.x	⊢ (𝜑 → 𝑋 ∈ (𝐴‘𝐺))
mptmpoopabbrd.y	⊢ (𝜑 → 𝑌 ∈ (𝐵‘𝐺))
mptmpoopabbrd.v	⊢ (𝜑 → {⟨𝑓, ℎ⟩ ∣ 𝜓} ∈ 𝑉)
mptmpoopabbrd.r	⊢ ((𝜑 ∧ 𝑓(𝐷‘𝐺)ℎ) → 𝜓)
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,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 mptmpoopabbrd

Step	Hyp	Ref	Expression
1		mptmpoopabbrd.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑊)
2		mptmpoopabbrd.m	. . . . 5 ⊢ 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴‘𝑔), 𝑏 ∈ (𝐵‘𝑔) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜒 ∧ 𝑓(𝐷‘𝑔)ℎ)}))
3		fveq2 6668	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝐴‘𝑔) = (𝐴‘𝐺))
4		fveq2 6668	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝐵‘𝑔) = (𝐵‘𝐺))
5		mptmpoopabbrd.2	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝜒 ↔ 𝜏))
6		fveq2 6668	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝐷‘𝑔) = (𝐷‘𝐺))
7	6	breqd 5038	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑓(𝐷‘𝑔)ℎ ↔ 𝑓(𝐷‘𝐺)ℎ))
8	5, 7	anbi12d 634	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((𝜒 ∧ 𝑓(𝐷‘𝑔)ℎ) ↔ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)))
9	8	opabbidv 5093	. . . . . 6 ⊢ (𝑔 = 𝐺 → {⟨𝑓, ℎ⟩ ∣ (𝜒 ∧ 𝑓(𝐷‘𝑔)ℎ)} = {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)})
10	3, 4, 9	mpoeq123dv 7237	. . . . 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 3415	. . . . . 6 ⊢ (𝐺 ∈ 𝑊 → 𝐺 ∈ V)
12	11	adantr 484	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐺 ∈ 𝑊) → 𝐺 ∈ V)
13		fvex 6681	. . . . . . 7 ⊢ (𝐴‘𝐺) ∈ V
14		fvex 6681	. . . . . . 7 ⊢ (𝐵‘𝐺) ∈ V
15	13, 14	pm3.2i 474	. . . . . 6 ⊢ ((𝐴‘𝐺) ∈ V ∧ (𝐵‘𝐺) ∈ V)
16		mpoexga 7794	. . . . . 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 6792	. . . 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 587	. . 3 ⊢ (𝜑 → (𝑀‘𝐺) = (𝑎 ∈ (𝐴‘𝐺), 𝑏 ∈ (𝐵‘𝐺) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)}))
20	19	oveqd 7181	. 2 ⊢ (𝜑 → (𝑋(𝑀‘𝐺)𝑌) = (𝑋(𝑎 ∈ (𝐴‘𝐺), 𝑏 ∈ (𝐵‘𝐺) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ)})𝑌))
21		mptmpoopabbrd.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐴‘𝐺))
22		mptmpoopabbrd.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝐵‘𝐺))
23		ancom 464	. . . . 5 ⊢ ((𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ) ↔ (𝑓(𝐷‘𝐺)ℎ ∧ 𝜃))
24	23	opabbii 5094	. . . 4 ⊢ {⟨𝑓, ℎ⟩ ∣ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)} = {⟨𝑓, ℎ⟩ ∣ (𝑓(𝐷‘𝐺)ℎ ∧ 𝜃)}
25		mptmpoopabbrd.r	. . . . 5 ⊢ ((𝜑 ∧ 𝑓(𝐷‘𝐺)ℎ) → 𝜓)
26		mptmpoopabbrd.v	. . . . 5 ⊢ (𝜑 → {⟨𝑓, ℎ⟩ ∣ 𝜓} ∈ 𝑉)
27	25, 26	opabresex2d 7216	. . . 4 ⊢ (𝜑 → {⟨𝑓, ℎ⟩ ∣ (𝑓(𝐷‘𝐺)ℎ ∧ 𝜃)} ∈ V)
28	24, 27	eqeltrid 2837	. . 3 ⊢ (𝜑 → {⟨𝑓, ℎ⟩ ∣ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)} ∈ V)
29		mptmpoopabbrd.1	. . . . . 6 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → (𝜏 ↔ 𝜃))
30	29	anbi1d 633	. . . . 5 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → ((𝜏 ∧ 𝑓(𝐷‘𝐺)ℎ) ↔ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)))
31	30	opabbidv 5093	. . . 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 7313	. . 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 1372	. 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 2773	1 ⊢ (𝜑 → (𝑋(𝑀‘𝐺)𝑌) = {⟨𝑓, ℎ⟩ ∣ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1542   ∈ wcel 2113  Vcvv 3397   class class class wbr 5027  {copab 5089   ↦ cmpt 5107  ‘cfv 6333  (class class class)co 7164   ∈ cmpo 7166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-ov 7167  df-oprab 7168  df-mpo 7169  df-1st 7707  df-2nd 7708

This theorem is referenced by:  mptmpoopabovd  7798  wlkson  27590