Theorem ovmpt3rab1 7678

Description: The value of an operation defined by the maps-to notation with a function into a class abstraction as a result. The domain of the function and the base set of the class abstraction may depend on the operands, using implicit substitution. (Contributed by AV, 16-Jul-2018.) (Revised by AV, 16-May-2019.)

Hypotheses

Ref	Expression
ovmpt3rab1.o	⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑧 ∈ 𝑀 ↦ {𝑎 ∈ 𝑁 ∣ 𝜑}))
ovmpt3rab1.m	⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑀 = 𝐾)
ovmpt3rab1.n	⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑁 = 𝐿)
ovmpt3rab1.p	⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ 𝜓))
ovmpt3rab1.x	⊢ Ⅎ𝑥𝜓
ovmpt3rab1.y	⊢ Ⅎ𝑦𝜓

Assertion

Ref	Expression
ovmpt3rab1	⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) → (𝑋𝑂𝑌) = (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓}))

Distinct variable groups:   𝑥,𝐾,𝑦,𝑧   𝐿,𝑎,𝑥,𝑦   𝑁,𝑎   𝑥,𝑉,𝑦   𝑥,𝑊,𝑦   𝑥,𝑈,𝑦   𝑋,𝑎,𝑥,𝑦,𝑧   𝑌,𝑎,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧, 𝑎)   𝜓(𝑥, 𝑦, 𝑧, 𝑎)   𝑈(𝑧, 𝑎)   𝐾(𝑎)   𝐿(𝑧)   𝑀(𝑥, 𝑦, 𝑧, 𝑎)   𝑁(𝑥, 𝑦, 𝑧)   𝑂(𝑥, 𝑦, 𝑧, 𝑎)   𝑉(𝑧, 𝑎)   𝑊(𝑧, 𝑎)

Proof of Theorem ovmpt3rab1

Step	Hyp	Ref	Expression
1		ovmpt3rab1.o	. . 3 ⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑧 ∈ 𝑀 ↦ {𝑎 ∈ 𝑁 ∣ 𝜑}))
2	1	a1i 11	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) → 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑧 ∈ 𝑀 ↦ {𝑎 ∈ 𝑁 ∣ 𝜑})))
3		ovmpt3rab1.m	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑀 = 𝐾)
4		ovmpt3rab1.n	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑁 = 𝐿)
5		ovmpt3rab1.p	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ 𝜓))
6	4, 5	rabeqbidv 3436	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → {𝑎 ∈ 𝑁 ∣ 𝜑} = {𝑎 ∈ 𝐿 ∣ 𝜓})
7	3, 6	mpteq12dv 5200	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑧 ∈ 𝑀 ↦ {𝑎 ∈ 𝑁 ∣ 𝜑}) = (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓}))
8	7	adantl 487	. 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑧 ∈ 𝑀 ↦ {𝑎 ∈ 𝑁 ∣ 𝜑}) = (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓}))
9		eqidd 2766	. 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) ∧ 𝑥 = 𝑋) → V = V)
10		elex 3478	. . 3 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
11	10	3ad2ant1 1151	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) → 𝑋 ∈ V)
12		elex 3478	. . 3 ⊢ (𝑌 ∈ 𝑊 → 𝑌 ∈ V)
13	12	3ad2ant2 1152	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) → 𝑌 ∈ V)
14		mptexg 7226	. . 3 ⊢ (𝐾 ∈ 𝑈 → (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓}) ∈ V)
15	14	3ad2ant3 1153	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) → (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓}) ∈ V)
16		nfv 1947	. 2 ⊢ Ⅎ𝑥(𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈)
17		nfv 1947	. 2 ⊢ Ⅎ𝑦(𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈)
18		nfcv 2927	. 2 ⊢ Ⅎ𝑦𝑋
19		nfcv 2927	. 2 ⊢ Ⅎ𝑥𝑌
20		nfcv 2927	. . 3 ⊢ Ⅎ𝑥𝐾
21		ovmpt3rab1.x	. . . 4 ⊢ Ⅎ𝑥𝜓
22		nfcv 2927	. . . 4 ⊢ Ⅎ𝑥𝐿
23	21, 22	nfrabw 3454	. . 3 ⊢ Ⅎ𝑥{𝑎 ∈ 𝐿 ∣ 𝜓}
24	20, 23	nfmpt 5211	. 2 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓})
25		nfcv 2927	. . 3 ⊢ Ⅎ𝑦𝐾
26		ovmpt3rab1.y	. . . 4 ⊢ Ⅎ𝑦𝜓
27		nfcv 2927	. . . 4 ⊢ Ⅎ𝑦𝐿
28	26, 27	nfrabw 3454	. . 3 ⊢ Ⅎ𝑦{𝑎 ∈ 𝐿 ∣ 𝜓}
29	25, 28	nfmpt 5211	. 2 ⊢ Ⅎ𝑦(𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓})
30	2, 8, 9, 11, 13, 15, 16, 17, 18, 19, 24, 29	ovmpodxf 7569	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) → (𝑋𝑂𝑌) = (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓}))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2146  {crab 3418  Vcvv 3457   ↦ cmpt 5194  (class class class)co 7419   ∈ cmpo 7421

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424

This theorem is used by:  ovmpt3rabdm  7679  elovmpt3rab1  7680