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Theorem ovmpt3rabdm 7153
Description: If the value of a function which is the result of an operation defined by the maps-to notation is not empty, the operands and the argument of the function must be sets. (Contributed by AV, 16-May-2019.)
Hypotheses
Ref Expression
ovmpt3rab1.o 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑧𝑀 ↦ {𝑎𝑁𝜑}))
ovmpt3rab1.m ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑀 = 𝐾)
ovmpt3rab1.n ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑁 = 𝐿)
Assertion
Ref Expression
ovmpt3rabdm (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ 𝐿𝑇) → dom (𝑋𝑂𝑌) = 𝐾)
Distinct variable groups:   𝑥,𝐾,𝑦,𝑧   𝐿,𝑎,𝑥,𝑦   𝑁,𝑎   𝑥,𝑉,𝑦   𝑥,𝑊,𝑦   𝑥,𝑈,𝑦   𝑋,𝑎,𝑥,𝑦,𝑧   𝑌,𝑎,𝑥,𝑦,𝑧   𝑧,𝐿   𝑧,𝑇   𝑧,𝑈   𝑧,𝑉   𝑧,𝑊
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑎)   𝑇(𝑥,𝑦,𝑎)   𝑈(𝑎)   𝐾(𝑎)   𝑀(𝑥,𝑦,𝑧,𝑎)   𝑁(𝑥,𝑦,𝑧)   𝑂(𝑥,𝑦,𝑧,𝑎)   𝑉(𝑎)   𝑊(𝑎)

Proof of Theorem ovmpt3rabdm
StepHypRef Expression
1 ovmpt3rab1.o . . . . 5 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑧𝑀 ↦ {𝑎𝑁𝜑}))
2 ovmpt3rab1.m . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑀 = 𝐾)
3 ovmpt3rab1.n . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑁 = 𝐿)
4 sbceq1a 3674 . . . . . 6 (𝑦 = 𝑌 → (𝜑[𝑌 / 𝑦]𝜑))
5 sbceq1a 3674 . . . . . 6 (𝑥 = 𝑋 → ([𝑌 / 𝑦]𝜑[𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
64, 5sylan9bbr 508 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝜑[𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
7 nfsbc1v 3683 . . . . 5 𝑥[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
8 nfcv 2970 . . . . . 6 𝑦𝑋
9 nfsbc1v 3683 . . . . . 6 𝑦[𝑌 / 𝑦]𝜑
108, 9nfsbc 3685 . . . . 5 𝑦[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
111, 2, 3, 6, 7, 10ovmpt3rab1 7152 . . . 4 ((𝑋𝑉𝑌𝑊𝐾𝑈) → (𝑋𝑂𝑌) = (𝑧𝐾 ↦ {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
1211adantr 474 . . 3 (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ 𝐿𝑇) → (𝑋𝑂𝑌) = (𝑧𝐾 ↦ {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
1312dmeqd 5559 . 2 (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ 𝐿𝑇) → dom (𝑋𝑂𝑌) = dom (𝑧𝐾 ↦ {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
14 rabexg 5037 . . . . 5 (𝐿𝑇 → {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
1514adantl 475 . . . 4 (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ 𝐿𝑇) → {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
1615ralrimivw 3177 . . 3 (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ 𝐿𝑇) → ∀𝑧𝐾 {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
17 dmmptg 5874 . . 3 (∀𝑧𝐾 {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V → dom (𝑧𝐾 ↦ {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}) = 𝐾)
1816, 17syl 17 . 2 (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ 𝐿𝑇) → dom (𝑧𝐾 ↦ {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}) = 𝐾)
1913, 18eqtrd 2862 1 (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ 𝐿𝑇) → dom (𝑋𝑂𝑌) = 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1113   = wceq 1658  wcel 2166  wral 3118  {crab 3122  Vcvv 3415  [wsbc 3663  cmpt 4953  dom cdm 5343  (class class class)co 6906  cmpt2 6908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pr 5128
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-ov 6909  df-oprab 6910  df-mpt2 6911
This theorem is referenced by:  elovmpt3rab1  7154
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