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Theorem mpoxneldm 8196
Description: If the first argument of an operation given by a maps-to rule is not an element of the first component of the domain or the second argument is not an element of the second component of the domain depending on the first argument, then the value of the operation is the empty set. (Contributed by AV, 25-Oct-2020.)
Hypothesis
Ref Expression
mpoxeldm.f 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
Assertion
Ref Expression
mpoxneldm ((𝑋𝐶𝑌𝑋 / 𝑥𝐷) → (𝑋𝐹𝑌) = ∅)
Distinct variable groups:   𝑥,𝐶,𝑦   𝑦,𝐷   𝑥,𝑋   𝑥,𝑌
Allowed substitution hints:   𝐷(𝑥)   𝑅(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑋(𝑦)   𝑌(𝑦)

Proof of Theorem mpoxneldm
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 df-nel 3047 . . . 4 (𝑋𝐶 ↔ ¬ 𝑋𝐶)
2 df-nel 3047 . . . 4 (𝑌𝑋 / 𝑥𝐷 ↔ ¬ 𝑌𝑋 / 𝑥𝐷)
31, 2orbi12i 913 . . 3 ((𝑋𝐶𝑌𝑋 / 𝑥𝐷) ↔ (¬ 𝑋𝐶 ∨ ¬ 𝑌𝑋 / 𝑥𝐷))
4 ianor 980 . . 3 (¬ (𝑋𝐶𝑌𝑋 / 𝑥𝐷) ↔ (¬ 𝑋𝐶 ∨ ¬ 𝑌𝑋 / 𝑥𝐷))
53, 4bitr4i 277 . 2 ((𝑋𝐶𝑌𝑋 / 𝑥𝐷) ↔ ¬ (𝑋𝐶𝑌𝑋 / 𝑥𝐷))
6 neq0 4345 . . . 4 (¬ (𝑋𝐹𝑌) = ∅ ↔ ∃𝑛 𝑛 ∈ (𝑋𝐹𝑌))
7 mpoxeldm.f . . . . . 6 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
87mpoxeldm 8195 . . . . 5 (𝑛 ∈ (𝑋𝐹𝑌) → (𝑋𝐶𝑌𝑋 / 𝑥𝐷))
98exlimiv 1933 . . . 4 (∃𝑛 𝑛 ∈ (𝑋𝐹𝑌) → (𝑋𝐶𝑌𝑋 / 𝑥𝐷))
106, 9sylbi 216 . . 3 (¬ (𝑋𝐹𝑌) = ∅ → (𝑋𝐶𝑌𝑋 / 𝑥𝐷))
1110con1i 147 . 2 (¬ (𝑋𝐶𝑌𝑋 / 𝑥𝐷) → (𝑋𝐹𝑌) = ∅)
125, 11sylbi 216 1 ((𝑋𝐶𝑌𝑋 / 𝑥𝐷) → (𝑋𝐹𝑌) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 845   = wceq 1541  wex 1781  wcel 2106  wnel 3046  csb 3893  c0 4322  (class class class)co 7408  cmpo 7410
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975
This theorem is referenced by:  nbgrnvtx0  28593
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