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Theorem mpoexw 8012
Description: Weak version of mpoex 8013 that holds without ax-rep 5243. 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 7481 . 2 Fun (𝑥𝐴, 𝑦𝐵𝐶)
3 mpoexw.4 . . . 4 𝑥𝐴𝑦𝐵 𝐶𝐷
41dmmpoga 8006 . . . 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 7688 . . 3 (𝐴 × 𝐵) ∈ V
95, 8eqeltri 2834 . 2 dom (𝑥𝐴, 𝑦𝐵𝐶) ∈ V
101rnmpo 7490 . . 3 ran (𝑥𝐴, 𝑦𝐵𝐶) = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶}
11 mpoexw.3 . . . 4 𝐷 ∈ V
123rspec 3234 . . . . . . . . 9 (𝑥𝐴 → ∀𝑦𝐵 𝐶𝐷)
1312r19.21bi 3235 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) → 𝐶𝐷)
14 eleq1a 2833 . . . . . . . 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 4028 . . . 4 {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶} ⊆ 𝐷
1911, 18ssexi 5280 . . 3 {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶} ∈ V
2010, 19eqeltri 2834 . 2 ran (𝑥𝐴, 𝑦𝐵𝐶) ∈ V
21 funexw 7885 . 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 397   = wceq 1542  wcel 2107  {cab 2714  wral 3065  wrex 3074  Vcvv 3446   × cxp 5632  dom cdm 5634  ran crn 5635  Fun wfun 6491  cmpo 7360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923
This theorem is referenced by:  prdsvallem  17337  prdsds  17347  plusffval  18504  grpsubfval  18795  mulgfval  18875  scaffval  20343  ipffval  21055
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