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Theorem mptexw 7978
Description: Weak version of mptex 7244 that holds without ax-rep 5278. If the domain and codomain of a function given by maps-to notation are sets, the function is a set. (Contributed by Rohan Ridenour, 13-Aug-2023.)
Hypotheses
Ref Expression
mptexw.1 𝐴 ∈ V
mptexw.2 𝐶 ∈ V
mptexw.3 𝑥𝐴 𝐵𝐶
Assertion
Ref Expression
mptexw (𝑥𝐴𝐵) ∈ V
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem mptexw
StepHypRef Expression
1 funmpt 6603 . 2 Fun (𝑥𝐴𝐵)
2 mptexw.1 . . 3 𝐴 ∈ V
3 eqid 2736 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43dmmptss 6260 . . 3 dom (𝑥𝐴𝐵) ⊆ 𝐴
52, 4ssexi 5321 . 2 dom (𝑥𝐴𝐵) ∈ V
6 mptexw.2 . . 3 𝐶 ∈ V
7 mptexw.3 . . . 4 𝑥𝐴 𝐵𝐶
83rnmptss 7142 . . . 4 (∀𝑥𝐴 𝐵𝐶 → ran (𝑥𝐴𝐵) ⊆ 𝐶)
97, 8ax-mp 5 . . 3 ran (𝑥𝐴𝐵) ⊆ 𝐶
106, 9ssexi 5321 . 2 ran (𝑥𝐴𝐵) ∈ V
11 funexw 7977 . 2 ((Fun (𝑥𝐴𝐵) ∧ dom (𝑥𝐴𝐵) ∈ V ∧ ran (𝑥𝐴𝐵) ∈ V) → (𝑥𝐴𝐵) ∈ V)
121, 5, 10, 11mp3an 1462 1 (𝑥𝐴𝐵) ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  wral 3060  Vcvv 3479  wss 3950  cmpt 5224  dom cdm 5684  ran crn 5685  Fun wfun 6554
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-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-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-fun 6562  df-fn 6563  df-f 6564
This theorem is referenced by:  grpinvfval  18997  odfval  19551
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