Theorem ovmpt3rab1 7708

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 3462	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → {𝑎 ∈ 𝑁 ∣ 𝜑} = {𝑎 ∈ 𝐿 ∣ 𝜓})
7	3, 6	mpteq12dv 5257	. . 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 2741	. 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) ∧ 𝑥 = 𝑋) → V = V)
10		elex 3509	. . 3 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
11	10	3ad2ant1 1133	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) → 𝑋 ∈ V)
12		elex 3509	. . 3 ⊢ (𝑌 ∈ 𝑊 → 𝑌 ∈ V)
13	12	3ad2ant2 1134	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) → 𝑌 ∈ V)
14		mptexg 7258	. . 3 ⊢ (𝐾 ∈ 𝑈 → (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓}) ∈ V)
15	14	3ad2ant3 1135	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) → (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓}) ∈ V)
16		nfv 1913	. 2 ⊢ Ⅎ𝑥(𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈)
17		nfv 1913	. 2 ⊢ Ⅎ𝑦(𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈)
18		nfcv 2908	. 2 ⊢ Ⅎ𝑦𝑋
19		nfcv 2908	. 2 ⊢ Ⅎ𝑥𝑌
20		nfcv 2908	. . 3 ⊢ Ⅎ𝑥𝐾
21		ovmpt3rab1.x	. . . 4 ⊢ Ⅎ𝑥𝜓
22		nfcv 2908	. . . 4 ⊢ Ⅎ𝑥𝐿
23	21, 22	nfrabw 3483	. . 3 ⊢ Ⅎ𝑥{𝑎 ∈ 𝐿 ∣ 𝜓}
24	20, 23	nfmpt 5273	. 2 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓})
25		nfcv 2908	. . 3 ⊢ Ⅎ𝑦𝐾
26		ovmpt3rab1.y	. . . 4 ⊢ Ⅎ𝑦𝜓
27		nfcv 2908	. . . 4 ⊢ Ⅎ𝑦𝐿
28	26, 27	nfrabw 3483	. . 3 ⊢ Ⅎ𝑦{𝑎 ∈ 𝐿 ∣ 𝜓}
29	25, 28	nfmpt 5273	. 2 ⊢ Ⅎ𝑦(𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓})
30	2, 8, 9, 11, 13, 15, 16, 17, 18, 19, 24, 29	ovmpodxf 7600	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) → (𝑋𝑂𝑌) = (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537  Ⅎwnf 1781   ∈ wcel 2108  {crab 3443  Vcvv 3488   ↦ cmpt 5249  (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-pr 5447

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-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

This theorem is referenced by:  ovmpt3rabdm  7709  elovmpt3rab1  7710