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Theorem mpoxeldm 8197
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 8051 . . 3 dom 𝐹 𝑥𝐶 ({𝑥} × 𝐷)
3 elfvdm 6922 . . . 4 (𝑁 ∈ (𝐹‘⟨𝑋, 𝑌⟩) → ⟨𝑋, 𝑌⟩ ∈ dom 𝐹)
4 df-ov 7408 . . . 4 (𝑋𝐹𝑌) = (𝐹‘⟨𝑋, 𝑌⟩)
53, 4eleq2s 2845 . . 3 (𝑁 ∈ (𝑋𝐹𝑌) → ⟨𝑋, 𝑌⟩ ∈ dom 𝐹)
62, 5sselid 3975 . 2 (𝑁 ∈ (𝑋𝐹𝑌) → ⟨𝑋, 𝑌⟩ ∈ 𝑥𝐶 ({𝑥} × 𝐷))
7 nfcsb1v 3913 . . 3 𝑥𝑋 / 𝑥𝐷
8 csbeq1a 3902 . . 3 (𝑥 = 𝑋𝐷 = 𝑋 / 𝑥𝐷)
97, 8opeliunxp2f 8196 . 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 395   = wceq 1533  wcel 2098  csb 3888  {csn 4623  cop 4629   ciun 4990   × cxp 5667  dom cdm 5669  cfv 6537  (class class class)co 7405  cmpo 7407
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975
This theorem is referenced by:  mpoxneldm  8198  nbgrcl  29100
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