Theorem mpoxeldm 8236

Description: If there is an element of the value of an operation given by a maps-to rule, then the first argument is an element of the first component of the domain and the second argument is an element of the second component of the domain depending on the first argument. (Contributed by AV, 25-Oct-2020.)

Hypothesis

Ref	Expression
mpoxeldm.f	⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)

Assertion

Ref	Expression
mpoxeldm	⊢ (𝑁 ∈ (𝑋𝐹𝑌) → (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷))

Distinct variable groups:   𝑥,𝐶,𝑦   𝑦,𝐷   𝑥,𝑋   𝑥,𝑌

Allowed substitution hints:   𝐷(𝑥)   𝑅(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑁(𝑥,𝑦)   𝑋(𝑦)   𝑌(𝑦)

Proof of Theorem mpoxeldm

Step	Hyp	Ref	Expression
1		mpoxeldm.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
2	1	dmmpossx 8091	. . 3 ⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ 𝐶 ({𝑥} × 𝐷)
3		elfvdm 6943	. . . 4 ⊢ (𝑁 ∈ (𝐹‘⟨𝑋, 𝑌⟩) → ⟨𝑋, 𝑌⟩ ∈ dom 𝐹)
4		df-ov 7434	. . . 4 ⊢ (𝑋𝐹𝑌) = (𝐹‘⟨𝑋, 𝑌⟩)
5	3, 4	eleq2s 2859	. . 3 ⊢ (𝑁 ∈ (𝑋𝐹𝑌) → ⟨𝑋, 𝑌⟩ ∈ dom 𝐹)
6	2, 5	sselid 3981	. 2 ⊢ (𝑁 ∈ (𝑋𝐹𝑌) → ⟨𝑋, 𝑌⟩ ∈ ∪ 𝑥 ∈ 𝐶 ({𝑥} × 𝐷))
7		nfcsb1v 3923	. . 3 ⊢ Ⅎ𝑥⦋𝑋 / 𝑥⦌𝐷
8		csbeq1a 3913	. . 3 ⊢ (𝑥 = 𝑋 → 𝐷 = ⦋𝑋 / 𝑥⦌𝐷)
9	7, 8	opeliunxp2f 8235	. 2 ⊢ (⟨𝑋, 𝑌⟩ ∈ ∪ 𝑥 ∈ 𝐶 ({𝑥} × 𝐷) ↔ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷))
10	6, 9	sylib 218	1 ⊢ (𝑁 ∈ (𝑋𝐹𝑌) → (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ⦋csb 3899  {csn 4626  ⟨cop 4632  ∪ ciun 4991   × cxp 5683  dom cdm 5685  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015

This theorem is referenced by:  mpoxneldm  8237  nbgrcl  29352  clnbgrcl  47808