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Theorem mpoxneldm 8147
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 914 . . 3 ((𝑋𝐶𝑌𝑋 / 𝑥𝐷) ↔ (¬ 𝑋𝐶 ∨ ¬ 𝑌𝑋 / 𝑥𝐷))
4 ianor 981 . . 3 (¬ (𝑋𝐶𝑌𝑋 / 𝑥𝐷) ↔ (¬ 𝑋𝐶 ∨ ¬ 𝑌𝑋 / 𝑥𝐷))
53, 4bitr4i 278 . 2 ((𝑋𝐶𝑌𝑋 / 𝑥𝐷) ↔ ¬ (𝑋𝐶𝑌𝑋 / 𝑥𝐷))
6 neq0 4309 . . . 4 (¬ (𝑋𝐹𝑌) = ∅ ↔ ∃𝑛 𝑛 ∈ (𝑋𝐹𝑌))
7 mpoxeldm.f . . . . . 6 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
87mpoxeldm 8146 . . . . 5 (𝑛 ∈ (𝑋𝐹𝑌) → (𝑋𝐶𝑌𝑋 / 𝑥𝐷))
98exlimiv 1934 . . . 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 397  wo 846   = wceq 1542  wex 1782  wcel 2107  wnel 3046  csb 3859  c0 4286  (class class class)co 7361  cmpo 7363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926
This theorem is referenced by:  nbgrnvtx0  28336
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