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Theorem mpoexw 7849
Description: Weak version of mpoex 7850 that holds without ax-rep 5179. 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 2737 . . 3 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐶)
21mpofun 7334 . 2 Fun (𝑥𝐴, 𝑦𝐵𝐶)
3 mpoexw.4 . . . 4 𝑥𝐴𝑦𝐵 𝐶𝐷
41dmmpoga 7843 . . . 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 7538 . . 3 (𝐴 × 𝐵) ∈ V
95, 8eqeltri 2834 . 2 dom (𝑥𝐴, 𝑦𝐵𝐶) ∈ V
101rnmpo 7343 . . 3 ran (𝑥𝐴, 𝑦𝐵𝐶) = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶}
11 mpoexw.3 . . . 4 𝐷 ∈ V
123rspec 3129 . . . . . . . . 9 (𝑥𝐴 → ∀𝑦𝐵 𝐶𝐷)
1312r19.21bi 3130 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) → 𝐶𝐷)
14 eleq1a 2833 . . . . . . . 8 (𝐶𝐷 → (𝑧 = 𝐶𝑧𝐷))
1513, 14syl 17 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → (𝑧 = 𝐶𝑧𝐷))
1615rexlimdva 3203 . . . . . 6 (𝑥𝐴 → (∃𝑦𝐵 𝑧 = 𝐶𝑧𝐷))
1716rexlimiv 3199 . . . . 5 (∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶𝑧𝐷)
1817abssi 3983 . . . 4 {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶} ⊆ 𝐷
1911, 18ssexi 5215 . . 3 {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶} ∈ V
2010, 19eqeltri 2834 . 2 ran (𝑥𝐴, 𝑦𝐵𝐶) ∈ V
21 funexw 7725 . 2 ((Fun (𝑥𝐴, 𝑦𝐵𝐶) ∧ dom (𝑥𝐴, 𝑦𝐵𝐶) ∈ V ∧ ran (𝑥𝐴, 𝑦𝐵𝐶) ∈ V) → (𝑥𝐴, 𝑦𝐵𝐶) ∈ V)
222, 9, 20, 21mp3an 1463 1 (𝑥𝐴, 𝑦𝐵𝐶) ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2110  {cab 2714  wral 3061  wrex 3062  Vcvv 3408   × cxp 5549  dom cdm 5551  ran crn 5552  Fun wfun 6374  cmpo 7215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fv 6388  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762
This theorem is referenced by:  prdsvallem  16959  prdsds  16969  plusffval  18120  grpsubfval  18411  mulgfval  18490  scaffval  19917  ipffval  20610
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