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Theorem mpoxeldm 8225
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 = (𝑥𝐶, 𝑦𝐷𝑅)
Assertion
Ref Expression
mpoxeldm (𝑁 ∈ (𝑋𝐹𝑌) → (𝑋𝐶𝑌𝑋 / 𝑥𝐷))
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
StepHypRef Expression
1 mpoxeldm.f . . . 4 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
21dmmpossx 8078 . . 3 dom 𝐹 𝑥𝐶 ({𝑥} × 𝐷)
3 elfvdm 6939 . . . 4 (𝑁 ∈ (𝐹‘⟨𝑋, 𝑌⟩) → ⟨𝑋, 𝑌⟩ ∈ dom 𝐹)
4 df-ov 7429 . . . 4 (𝑋𝐹𝑌) = (𝐹‘⟨𝑋, 𝑌⟩)
53, 4eleq2s 2847 . . 3 (𝑁 ∈ (𝑋𝐹𝑌) → ⟨𝑋, 𝑌⟩ ∈ dom 𝐹)
62, 5sselid 3980 . 2 (𝑁 ∈ (𝑋𝐹𝑌) → ⟨𝑋, 𝑌⟩ ∈ 𝑥𝐶 ({𝑥} × 𝐷))
7 nfcsb1v 3919 . . 3 𝑥𝑋 / 𝑥𝐷
8 csbeq1a 3908 . . 3 (𝑥 = 𝑋𝐷 = 𝑋 / 𝑥𝐷)
97, 8opeliunxp2f 8224 . 2 (⟨𝑋, 𝑌⟩ ∈ 𝑥𝐶 ({𝑥} × 𝐷) ↔ (𝑋𝐶𝑌𝑋 / 𝑥𝐷))
106, 9sylib 217 1 (𝑁 ∈ (𝑋𝐹𝑌) → (𝑋𝐶𝑌𝑋 / 𝑥𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  csb 3894  {csn 4632  cop 4638   ciun 5000   × cxp 5680  dom cdm 5682  cfv 6553  (class class class)co 7426  cmpo 7428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 8001  df-2nd 8002
This theorem is referenced by:  mpoxneldm  8226  nbgrcl  29176  clnbgrcl  47208
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