Theorem ovmpt3rab1 7626

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 3419	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → {𝑎 ∈ 𝑁 ∣ 𝜑} = {𝑎 ∈ 𝐿 ∣ 𝜓})
7	3, 6	mpteq12dv 5187	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑧 ∈ 𝑀 ↦ {𝑎 ∈ 𝑁 ∣ 𝜑}) = (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓}))
8	7	adantl 481	. 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑧 ∈ 𝑀 ↦ {𝑎 ∈ 𝑁 ∣ 𝜑}) = (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓}))
9		eqidd 2738	. 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) ∧ 𝑥 = 𝑋) → V = V)
10		elex 3463	. . 3 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
11	10	3ad2ant1 1134	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) → 𝑋 ∈ V)
12		elex 3463	. . 3 ⊢ (𝑌 ∈ 𝑊 → 𝑌 ∈ V)
13	12	3ad2ant2 1135	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) → 𝑌 ∈ V)
14		mptexg 7177	. . 3 ⊢ (𝐾 ∈ 𝑈 → (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓}) ∈ V)
15	14	3ad2ant3 1136	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) → (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓}) ∈ V)
16		nfv 1916	. 2 ⊢ Ⅎ𝑥(𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈)
17		nfv 1916	. 2 ⊢ Ⅎ𝑦(𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈)
18		nfcv 2899	. 2 ⊢ Ⅎ𝑦𝑋
19		nfcv 2899	. 2 ⊢ Ⅎ𝑥𝑌
20		nfcv 2899	. . 3 ⊢ Ⅎ𝑥𝐾
21		ovmpt3rab1.x	. . . 4 ⊢ Ⅎ𝑥𝜓
22		nfcv 2899	. . . 4 ⊢ Ⅎ𝑥𝐿
23	21, 22	nfrabw 3438	. . 3 ⊢ Ⅎ𝑥{𝑎 ∈ 𝐿 ∣ 𝜓}
24	20, 23	nfmpt 5198	. 2 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓})
25		nfcv 2899	. . 3 ⊢ Ⅎ𝑦𝐾
26		ovmpt3rab1.y	. . . 4 ⊢ Ⅎ𝑦𝜓
27		nfcv 2899	. . . 4 ⊢ Ⅎ𝑦𝐿
28	26, 27	nfrabw 3438	. . 3 ⊢ Ⅎ𝑦{𝑎 ∈ 𝐿 ∣ 𝜓}
29	25, 28	nfmpt 5198	. 2 ⊢ Ⅎ𝑦(𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓})
30	2, 8, 9, 11, 13, 15, 16, 17, 18, 19, 24, 29	ovmpodxf 7518	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) → (𝑋𝑂𝑌) = (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  Ⅎwnf 1785   ∈ wcel 2114  {crab 3401  Vcvv 3442   ↦ cmpt 5181  (class class class)co 7368   ∈ cmpo 7370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373

This theorem is referenced by:  ovmpt3rabdm  7627  elovmpt3rab1  7628