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Theorem mpt2xneldm 7570
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
mpt2xeldm2.f 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
Assertion
Ref Expression
mpt2xneldm ((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 mpt2xneldm
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 df-nel 3081 . . . 4 (𝑋𝐶 ↔ ¬ 𝑋𝐶)
2 df-nel 3081 . . . 4 (𝑌𝑋 / 𝑥𝐷 ↔ ¬ 𝑌𝑋 / 𝑥𝐷)
31, 2orbi12i 929 . . 3 ((𝑋𝐶𝑌𝑋 / 𝑥𝐷) ↔ (¬ 𝑋𝐶 ∨ ¬ 𝑌𝑋 / 𝑥𝐷))
4 ianor 995 . . 3 (¬ (𝑋𝐶𝑌𝑋 / 𝑥𝐷) ↔ (¬ 𝑋𝐶 ∨ ¬ 𝑌𝑋 / 𝑥𝐷))
53, 4bitr4i 269 . 2 ((𝑋𝐶𝑌𝑋 / 𝑥𝐷) ↔ ¬ (𝑋𝐶𝑌𝑋 / 𝑥𝐷))
6 neq0 4128 . . . 4 (¬ (𝑋𝐹𝑌) = ∅ ↔ ∃𝑛 𝑛 ∈ (𝑋𝐹𝑌))
7 mpt2xeldm2.f . . . . . 6 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
87mpt2xeldm 7569 . . . . 5 (𝑛 ∈ (𝑋𝐹𝑌) → (𝑋𝐶𝑌𝑋 / 𝑥𝐷))
98exlimiv 2023 . . . 4 (∃𝑛 𝑛 ∈ (𝑋𝐹𝑌) → (𝑋𝐶𝑌𝑋 / 𝑥𝐷))
106, 9sylbi 208 . . 3 (¬ (𝑋𝐹𝑌) = ∅ → (𝑋𝐶𝑌𝑋 / 𝑥𝐷))
1110con1i 146 . 2 (¬ (𝑋𝐶𝑌𝑋 / 𝑥𝐷) → (𝑋𝐹𝑌) = ∅)
125, 11sylbi 208 1 ((𝑋𝐶𝑌𝑋 / 𝑥𝐷) → (𝑋𝐹𝑌) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wo 865   = wceq 1637  wex 1859  wcel 2158  wnel 3080  csb 3725  c0 4113  (class class class)co 6871  cmpt2 6873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-8 2160  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-sep 4971  ax-nul 4980  ax-pow 5032  ax-pr 5093  ax-un 7176
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ne 2978  df-nel 3081  df-ral 3100  df-rex 3101  df-rab 3104  df-v 3392  df-sbc 3631  df-csb 3726  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-nul 4114  df-if 4277  df-sn 4368  df-pr 4370  df-op 4374  df-uni 4627  df-iun 4710  df-br 4841  df-opab 4903  df-mpt 4920  df-id 5216  df-xp 5314  df-rel 5315  df-cnv 5316  df-co 5317  df-dm 5318  df-rn 5319  df-res 5320  df-ima 5321  df-iota 6061  df-fun 6100  df-fv 6106  df-ov 6874  df-oprab 6875  df-mpt2 6876  df-1st 7395  df-2nd 7396
This theorem is referenced by:  nbgrnvtx0  26444
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