Theorem mpoxeldm 8058

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 7938	. . 3 ⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ 𝐶 ({𝑥} × 𝐷)
3		elfvdm 6838	. . . 4 ⊢ (𝑁 ∈ (𝐹‘⟨𝑋, 𝑌⟩) → ⟨𝑋, 𝑌⟩ ∈ dom 𝐹)
4		df-ov 7310	. . . 4 ⊢ (𝑋𝐹𝑌) = (𝐹‘⟨𝑋, 𝑌⟩)
5	3, 4	eleq2s 2855	. . 3 ⊢ (𝑁 ∈ (𝑋𝐹𝑌) → ⟨𝑋, 𝑌⟩ ∈ dom 𝐹)
6	2, 5	sselid 3924	. 2 ⊢ (𝑁 ∈ (𝑋𝐹𝑌) → ⟨𝑋, 𝑌⟩ ∈ ∪ 𝑥 ∈ 𝐶 ({𝑥} × 𝐷))
7		nfcsb1v 3862	. . 3 ⊢ Ⅎ𝑥⦋𝑋 / 𝑥⦌𝐷
8		csbeq1a 3851	. . 3 ⊢ (𝑥 = 𝑋 → 𝐷 = ⦋𝑋 / 𝑥⦌𝐷)
9	7, 8	opeliunxp2f 8057	. 2 ⊢ (⟨𝑋, 𝑌⟩ ∈ ∪ 𝑥 ∈ 𝐶 ({𝑥} × 𝐷) ↔ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷))
10	6, 9	sylib 217	1 ⊢ (𝑁 ∈ (𝑋𝐹𝑌) → (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1539   ∈ wcel 2104  ⦋csb 3837  {csn 4565  ⟨cop 4571  ∪ ciun 4931   × cxp 5598  dom cdm 5600  ‘cfv 6458  (class class class)co 7307   ∈ cmpo 7309

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361  ax-un 7620

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ral 3063  df-rex 3072  df-rab 3306  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-iota 6410  df-fun 6460  df-fv 6466  df-ov 7310  df-oprab 7311  df-mpo 7312  df-1st 7863  df-2nd 7864

This theorem is referenced by:  mpoxneldm  8059  nbgrcl  27751