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Theorem mpoexw 8104
Description: Weak version of mpoex 8105 that holds without ax-rep 5278. 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 2736 . . 3 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐶)
21mpofun 7558 . 2 Fun (𝑥𝐴, 𝑦𝐵𝐶)
3 mpoexw.4 . . . 4 𝑥𝐴𝑦𝐵 𝐶𝐷
41dmmpoga 8099 . . . 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 7774 . . 3 (𝐴 × 𝐵) ∈ V
95, 8eqeltri 2836 . 2 dom (𝑥𝐴, 𝑦𝐵𝐶) ∈ V
101rnmpo 7567 . . 3 ran (𝑥𝐴, 𝑦𝐵𝐶) = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶}
11 mpoexw.3 . . . 4 𝐷 ∈ V
123rspec 3249 . . . . . . . . 9 (𝑥𝐴 → ∀𝑦𝐵 𝐶𝐷)
1312r19.21bi 3250 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) → 𝐶𝐷)
14 eleq1a 2835 . . . . . . . 8 (𝐶𝐷 → (𝑧 = 𝐶𝑧𝐷))
1513, 14syl 17 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → (𝑧 = 𝐶𝑧𝐷))
1615rexlimdva 3154 . . . . . 6 (𝑥𝐴 → (∃𝑦𝐵 𝑧 = 𝐶𝑧𝐷))
1716rexlimiv 3147 . . . . 5 (∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶𝑧𝐷)
1817abssi 4069 . . . 4 {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶} ⊆ 𝐷
1911, 18ssexi 5321 . . 3 {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶} ∈ V
2010, 19eqeltri 2836 . 2 ran (𝑥𝐴, 𝑦𝐵𝐶) ∈ V
21 funexw 7977 . 2 ((Fun (𝑥𝐴, 𝑦𝐵𝐶) ∧ dom (𝑥𝐴, 𝑦𝐵𝐶) ∈ V ∧ ran (𝑥𝐴, 𝑦𝐵𝐶) ∈ V) → (𝑥𝐴, 𝑦𝐵𝐶) ∈ V)
222, 9, 20, 21mp3an 1462 1 (𝑥𝐴, 𝑦𝐵𝐶) ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  {cab 2713  wral 3060  wrex 3069  Vcvv 3479   × cxp 5682  dom cdm 5684  ran crn 5685  Fun wfun 6554  cmpo 7434
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-pow 5364  ax-pr 5431  ax-un 7756
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-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  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-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016
This theorem is referenced by:  mptmpoopabbrd  8106  prdsvallem  17500  prdsds  17510  plusffval  18660  grpsubfval  19002  mulgfval  19088  scaffval  20879  ipffval  21667
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