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Theorem mpoexw 8102
Description: Weak version of mpoex 8103 that holds without ax-rep 5285. If the domain and codomain of an operation given by maps-to notation are sets, the operation is a set. (Contributed by Rohan Ridenour, 14-Aug-2023.)
Hypotheses
Ref Expression
mpoexw.1 𝐴 ∈ V
mpoexw.2 𝐵 ∈ V
mpoexw.3 𝐷 ∈ V
mpoexw.4 𝑥𝐴𝑦𝐵 𝐶𝐷
Assertion
Ref Expression
mpoexw (𝑥𝐴, 𝑦𝐵𝐶) ∈ V
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐷,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)

Proof of Theorem mpoexw
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eqid 2735 . . 3 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐶)
21mpofun 7557 . 2 Fun (𝑥𝐴, 𝑦𝐵𝐶)
3 mpoexw.4 . . . 4 𝑥𝐴𝑦𝐵 𝐶𝐷
41dmmpoga 8097 . . . 4 (∀𝑥𝐴𝑦𝐵 𝐶𝐷 → dom (𝑥𝐴, 𝑦𝐵𝐶) = (𝐴 × 𝐵))
53, 4ax-mp 5 . . 3 dom (𝑥𝐴, 𝑦𝐵𝐶) = (𝐴 × 𝐵)
6 mpoexw.1 . . . 4 𝐴 ∈ V
7 mpoexw.2 . . . 4 𝐵 ∈ V
86, 7xpex 7772 . . 3 (𝐴 × 𝐵) ∈ V
95, 8eqeltri 2835 . 2 dom (𝑥𝐴, 𝑦𝐵𝐶) ∈ V
101rnmpo 7566 . . 3 ran (𝑥𝐴, 𝑦𝐵𝐶) = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶}
11 mpoexw.3 . . . 4 𝐷 ∈ V
123rspec 3248 . . . . . . . . 9 (𝑥𝐴 → ∀𝑦𝐵 𝐶𝐷)
1312r19.21bi 3249 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) → 𝐶𝐷)
14 eleq1a 2834 . . . . . . . 8 (𝐶𝐷 → (𝑧 = 𝐶𝑧𝐷))
1513, 14syl 17 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → (𝑧 = 𝐶𝑧𝐷))
1615rexlimdva 3153 . . . . . 6 (𝑥𝐴 → (∃𝑦𝐵 𝑧 = 𝐶𝑧𝐷))
1716rexlimiv 3146 . . . . 5 (∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶𝑧𝐷)
1817abssi 4080 . . . 4 {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶} ⊆ 𝐷
1911, 18ssexi 5328 . . 3 {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶} ∈ V
2010, 19eqeltri 2835 . 2 ran (𝑥𝐴, 𝑦𝐵𝐶) ∈ V
21 funexw 7975 . 2 ((Fun (𝑥𝐴, 𝑦𝐵𝐶) ∧ dom (𝑥𝐴, 𝑦𝐵𝐶) ∈ V ∧ ran (𝑥𝐴, 𝑦𝐵𝐶) ∈ V) → (𝑥𝐴, 𝑦𝐵𝐶) ∈ V)
222, 9, 20, 21mp3an 1460 1 (𝑥𝐴, 𝑦𝐵𝐶) ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  Vcvv 3478   × cxp 5687  dom cdm 5689  ran crn 5690  Fun wfun 6557  cmpo 7433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014
This theorem is referenced by:  mptmpoopabbrd  8104  prdsvallem  17501  prdsds  17511  plusffval  18672  grpsubfval  19014  mulgfval  19100  scaffval  20895  ipffval  21684
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