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Theorem mpoxneldm 8218
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 3044 . . . 4 (𝑋𝐶 ↔ ¬ 𝑋𝐶)
2 df-nel 3044 . . . 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 278 . 2 ((𝑋𝐶𝑌𝑋 / 𝑥𝐷) ↔ ¬ (𝑋𝐶𝑌𝑋 / 𝑥𝐷))
6 neq0 4346 . . . 4 (¬ (𝑋𝐹𝑌) = ∅ ↔ ∃𝑛 𝑛 ∈ (𝑋𝐹𝑌))
7 mpoxeldm.f . . . . . 6 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
87mpoxeldm 8217 . . . . 5 (𝑛 ∈ (𝑋𝐹𝑌) → (𝑋𝐶𝑌𝑋 / 𝑥𝐷))
98exlimiv 1926 . . . 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 395  wo 846   = wceq 1534  wex 1774  wcel 2099  wnel 3043  csb 3892  c0 4323  (class class class)co 7420  cmpo 7422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-nel 3044  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994
This theorem is referenced by:  nbgrnvtx0  29165
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