Theorem mpt2xeldm 7224

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
mpt2xeldm2.f	⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)

Assertion

Ref	Expression
mpt2xeldm	⊢ (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 mpt2xeldm

Step	Hyp	Ref	Expression
1		mpt2xeldm2.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
2	1	dmmpt2ssx 7124	. . 3 ⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ 𝐶 ({𝑥} × 𝐷)
3		elfvdm 6130	. . . 4 ⊢ (𝑁 ∈ (𝐹‘⟨𝑋, 𝑌⟩) → ⟨𝑋, 𝑌⟩ ∈ dom 𝐹)
4		df-ov 6552	. . . 4 ⊢ (𝑋𝐹𝑌) = (𝐹‘⟨𝑋, 𝑌⟩)
5	3, 4	eleq2s 2706	. . 3 ⊢ (𝑁 ∈ (𝑋𝐹𝑌) → ⟨𝑋, 𝑌⟩ ∈ dom 𝐹)
6	2, 5	sseldi 3566	. 2 ⊢ (𝑁 ∈ (𝑋𝐹𝑌) → ⟨𝑋, 𝑌⟩ ∈ ∪ 𝑥 ∈ 𝐶 ({𝑥} × 𝐷))
7		nfcsb1v 3515	. . 3 ⊢ Ⅎ𝑥⦋𝑋 / 𝑥⦌𝐷
8		csbeq1a 3508	. . 3 ⊢ (𝑥 = 𝑋 → 𝐷 = ⦋𝑋 / 𝑥⦌𝐷)
9	7, 8	opeliunxp2f 7223	. 2 ⊢ (⟨𝑋, 𝑌⟩ ∈ ∪ 𝑥 ∈ 𝐶 ({𝑥} × 𝐷) ↔ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷))
10	6, 9	sylib 207	1 ⊢ (𝑁 ∈ (𝑋𝐹𝑌) → (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ⦋csb 3499  {csn 4125  ⟨cop 4131  ∪ ciun 4455   × cxp 5036  dom cdm 5038  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060

This theorem is referenced by:  mpt2xneldm  7225  nbgrcl  40559