Theorem ovmpt3rabdm 7528

Description: If the value of a function which is the result of an operation defined by the maps-to notation is not empty, the operands and the argument of the function must be sets. (Contributed 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)

Assertion

Ref	Expression
ovmpt3rabdm	⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) ∧ 𝐿 ∈ 𝑇) → dom (𝑋𝑂𝑌) = 𝐾)

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   𝑧,𝐿   𝑧,𝑇   𝑧,𝑈   𝑧,𝑉   𝑧,𝑊

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

Proof of Theorem ovmpt3rabdm

Step	Hyp	Ref	Expression
1		ovmpt3rab1.o	. . . . 5 ⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑧 ∈ 𝑀 ↦ {𝑎 ∈ 𝑁 ∣ 𝜑}))
2		ovmpt3rab1.m	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑀 = 𝐾)
3		ovmpt3rab1.n	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑁 = 𝐿)
4		sbceq1a 3727	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝜑 ↔ [𝑌 / 𝑦]𝜑))
5		sbceq1a 3727	. . . . . 6 ⊢ (𝑥 = 𝑋 → ([𝑌 / 𝑦]𝜑 ↔ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
6	4, 5	sylan9bbr 511	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
7		nfsbc1v 3736	. . . . 5 ⊢ Ⅎ𝑥[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
8		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑦𝑋
9		nfsbc1v 3736	. . . . . 6 ⊢ Ⅎ𝑦[𝑌 / 𝑦]𝜑
10	8, 9	nfsbcw 3738	. . . . 5 ⊢ Ⅎ𝑦[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
11	1, 2, 3, 6, 7, 10	ovmpt3rab1 7527	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) → (𝑋𝑂𝑌) = (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
12	11	adantr 481	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) ∧ 𝐿 ∈ 𝑇) → (𝑋𝑂𝑌) = (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
13	12	dmeqd 5814	. 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) ∧ 𝐿 ∈ 𝑇) → dom (𝑋𝑂𝑌) = dom (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
14		rabexg 5255	. . . . 5 ⊢ (𝐿 ∈ 𝑇 → {𝑎 ∈ 𝐿 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
15	14	adantl 482	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) ∧ 𝐿 ∈ 𝑇) → {𝑎 ∈ 𝐿 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
16	15	ralrimivw 3104	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) ∧ 𝐿 ∈ 𝑇) → ∀𝑧 ∈ 𝐾 {𝑎 ∈ 𝐿 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
17		dmmptg 6145	. . 3 ⊢ (∀𝑧 ∈ 𝐾 {𝑎 ∈ 𝐿 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V → dom (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑}) = 𝐾)
18	16, 17	syl 17	. 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) ∧ 𝐿 ∈ 𝑇) → dom (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑}) = 𝐾)
19	13, 18	eqtrd 2778	1 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) ∧ 𝐿 ∈ 𝑇) → dom (𝑋𝑂𝑌) = 𝐾)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {crab 3068  Vcvv 3432  [wsbc 3716   ↦ cmpt 5157  dom cdm 5589  (class class class)co 7275   ∈ cmpo 7277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280

This theorem is referenced by:  elovmpt3rab1  7529