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Theorem of0r 32689
Description: Function operation with the empty function. (Contributed by Thierry Arnoux, 27-May-2025.)
Assertion
Ref Expression
of0r (𝐹f 𝑅∅) = ∅

Proof of Theorem of0r
Dummy variables 𝑓 𝑔 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-of 7698 . . . 4 f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))))
21a1i 11 . . 3 (𝐹 ∈ V → ∘f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥)))))
3 dmeq 5913 . . . . . . 7 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
4 dmeq 5913 . . . . . . 7 (𝑔 = ∅ → dom 𝑔 = dom ∅)
53, 4ineqan12d 4221 . . . . . 6 ((𝑓 = 𝐹𝑔 = ∅) → (dom 𝑓 ∩ dom 𝑔) = (dom 𝐹 ∩ dom ∅))
65mpteq1d 5236 . . . . 5 ((𝑓 = 𝐹𝑔 = ∅) → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom ∅) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))))
76adantl 481 . . . 4 ((𝐹 ∈ V ∧ (𝑓 = 𝐹𝑔 = ∅)) → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom ∅) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))))
8 dm0 5930 . . . . . . . 8 dom ∅ = ∅
98ineq2i 4216 . . . . . . 7 (dom 𝐹 ∩ dom ∅) = (dom 𝐹 ∩ ∅)
10 in0 4394 . . . . . . 7 (dom 𝐹 ∩ ∅) = ∅
119, 10eqtri 2764 . . . . . 6 (dom 𝐹 ∩ dom ∅) = ∅
1211a1i 11 . . . . 5 ((𝐹 ∈ V ∧ (𝑓 = 𝐹𝑔 = ∅)) → (dom 𝐹 ∩ dom ∅) = ∅)
1312mpteq1d 5236 . . . 4 ((𝐹 ∈ V ∧ (𝑓 = 𝐹𝑔 = ∅)) → (𝑥 ∈ (dom 𝐹 ∩ dom ∅) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))) = (𝑥 ∈ ∅ ↦ ((𝑓𝑥)𝑅(𝑔𝑥))))
14 mpt0 6709 . . . . 5 (𝑥 ∈ ∅ ↦ ((𝑓𝑥)𝑅(𝑔𝑥))) = ∅
1514a1i 11 . . . 4 ((𝐹 ∈ V ∧ (𝑓 = 𝐹𝑔 = ∅)) → (𝑥 ∈ ∅ ↦ ((𝑓𝑥)𝑅(𝑔𝑥))) = ∅)
167, 13, 153eqtrd 2780 . . 3 ((𝐹 ∈ V ∧ (𝑓 = 𝐹𝑔 = ∅)) → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))) = ∅)
17 id 22 . . 3 (𝐹 ∈ V → 𝐹 ∈ V)
18 0ex 5306 . . . 4 ∅ ∈ V
1918a1i 11 . . 3 (𝐹 ∈ V → ∅ ∈ V)
202, 16, 17, 19, 19ovmpod 7586 . 2 (𝐹 ∈ V → (𝐹f 𝑅∅) = ∅)
211reldmmpo 7568 . . 3 Rel dom ∘f 𝑅
2221ovprc1 7471 . 2 𝐹 ∈ V → (𝐹f 𝑅∅) = ∅)
2320, 22pm2.61i 182 1 (𝐹f 𝑅∅) = ∅
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1539  wcel 2107  Vcvv 3479  cin 3949  c0 4332  cmpt 5224  dom cdm 5684  cfv 6560  (class class class)co 7432  cmpo 7434  f cof 7696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fn 6563  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698
This theorem is referenced by:  1arithidom  33566
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