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Theorem mpoxeldm 7864
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 7750 . . 3 dom 𝐹 𝑥𝐶 ({𝑥} × 𝐷)
3 elfvdm 6681 . . . 4 (𝑁 ∈ (𝐹‘⟨𝑋, 𝑌⟩) → ⟨𝑋, 𝑌⟩ ∈ dom 𝐹)
4 df-ov 7142 . . . 4 (𝑋𝐹𝑌) = (𝐹‘⟨𝑋, 𝑌⟩)
53, 4eleq2s 2911 . . 3 (𝑁 ∈ (𝑋𝐹𝑌) → ⟨𝑋, 𝑌⟩ ∈ dom 𝐹)
62, 5sseldi 3916 . 2 (𝑁 ∈ (𝑋𝐹𝑌) → ⟨𝑋, 𝑌⟩ ∈ 𝑥𝐶 ({𝑥} × 𝐷))
7 nfcsb1v 3855 . . 3 𝑥𝑋 / 𝑥𝐷
8 csbeq1a 3845 . . 3 (𝑥 = 𝑋𝐷 = 𝑋 / 𝑥𝐷)
97, 8opeliunxp2f 7863 . 2 (⟨𝑋, 𝑌⟩ ∈ 𝑥𝐶 ({𝑥} × 𝐷) ↔ (𝑋𝐶𝑌𝑋 / 𝑥𝐷))
106, 9sylib 221 1 (𝑁 ∈ (𝑋𝐹𝑌) → (𝑋𝐶𝑌𝑋 / 𝑥𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  csb 3831  {csn 4528  cop 4534   ciun 4884   × cxp 5521  dom cdm 5523  cfv 6328  (class class class)co 7139  cmpo 7141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676
This theorem is referenced by:  mpoxneldm  7865  nbgrcl  27128
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