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Theorem mpoexw 7762
 Description: Weak version of mpoex 7763 that holds without ax-rep 5155. 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 2798 . . 3 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐶)
21mpofun 7256 . 2 Fun (𝑥𝐴, 𝑦𝐵𝐶)
3 mpoexw.4 . . . 4 𝑥𝐴𝑦𝐵 𝐶𝐷
41dmmpoga 7756 . . . 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 7459 . . 3 (𝐴 × 𝐵) ∈ V
95, 8eqeltri 2886 . 2 dom (𝑥𝐴, 𝑦𝐵𝐶) ∈ V
101rnmpo 7265 . . 3 ran (𝑥𝐴, 𝑦𝐵𝐶) = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶}
11 mpoexw.3 . . . 4 𝐷 ∈ V
123rspec 3172 . . . . . . . . 9 (𝑥𝐴 → ∀𝑦𝐵 𝐶𝐷)
1312r19.21bi 3173 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) → 𝐶𝐷)
14 eleq1a 2885 . . . . . . . 8 (𝐶𝐷 → (𝑧 = 𝐶𝑧𝐷))
1513, 14syl 17 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → (𝑧 = 𝐶𝑧𝐷))
1615rexlimdva 3243 . . . . . 6 (𝑥𝐴 → (∃𝑦𝐵 𝑧 = 𝐶𝑧𝐷))
1716rexlimiv 3239 . . . . 5 (∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶𝑧𝐷)
1817abssi 3997 . . . 4 {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶} ⊆ 𝐷
1911, 18ssexi 5191 . . 3 {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶} ∈ V
2010, 19eqeltri 2886 . 2 ran (𝑥𝐴, 𝑦𝐵𝐶) ∈ V
21 funexw 7638 . 2 ((Fun (𝑥𝐴, 𝑦𝐵𝐶) ∧ dom (𝑥𝐴, 𝑦𝐵𝐶) ∈ V ∧ ran (𝑥𝐴, 𝑦𝐵𝐶) ∈ V) → (𝑥𝐴, 𝑦𝐵𝐶) ∈ V)
222, 9, 20, 21mp3an 1458 1 (𝑥𝐴, 𝑦𝐵𝐶) ∈ V
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2776  ∀wral 3106  ∃wrex 3107  Vcvv 3441   × cxp 5518  dom cdm 5520  ran crn 5521  Fun wfun 6319   ∈ cmpo 7138 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-fv 6333  df-oprab 7140  df-mpo 7141  df-1st 7674  df-2nd 7675 This theorem is referenced by:  prdsds  16732  plusffval  17853  grpsubfval  18143  mulgfval  18222  scaffval  19649  ipffval  20342
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