Theorem ovmpt3rabdm 7627

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 3753	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝜑 ↔ [𝑌 / 𝑦]𝜑))
5		sbceq1a 3753	. . . . . 6 ⊢ (𝑥 = 𝑋 → ([𝑌 / 𝑦]𝜑 ↔ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
6	4, 5	sylan9bbr 510	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
7		nfsbc1v 3762	. . . . 5 ⊢ Ⅎ𝑥[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
8		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑦𝑋
9		nfsbc1v 3762	. . . . . 6 ⊢ Ⅎ𝑦[𝑌 / 𝑦]𝜑
10	8, 9	nfsbcw 3764	. . . . 5 ⊢ Ⅎ𝑦[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
11	1, 2, 3, 6, 7, 10	ovmpt3rab1 7626	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) → (𝑋𝑂𝑌) = (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
12	11	adantr 480	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) ∧ 𝐿 ∈ 𝑇) → (𝑋𝑂𝑌) = (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
13	12	dmeqd 5862	. 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) ∧ 𝐿 ∈ 𝑇) → dom (𝑋𝑂𝑌) = dom (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
14		rabexg 5284	. . . . 5 ⊢ (𝐿 ∈ 𝑇 → {𝑎 ∈ 𝐿 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
15	14	adantl 481	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) ∧ 𝐿 ∈ 𝑇) → {𝑎 ∈ 𝐿 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
16	15	ralrimivw 3134	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) ∧ 𝐿 ∈ 𝑇) → ∀𝑧 ∈ 𝐾 {𝑎 ∈ 𝐿 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
17		dmmptg 6208	. . 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 2772	1 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) ∧ 𝐿 ∈ 𝑇) → dom (𝑋𝑂𝑌) = 𝐾)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3401  Vcvv 3442  [wsbc 3742   ↦ cmpt 5181  dom cdm 5632  (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-pw 4558  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:  elovmpt3rab1  7628