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Theorem mptexw 7976
Description: Weak version of mptex 7243 that holds without ax-rep 5285. 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 6606 . 2 Fun (𝑥𝐴𝐵)
2 mptexw.1 . . 3 𝐴 ∈ V
3 eqid 2735 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43dmmptss 6263 . . 3 dom (𝑥𝐴𝐵) ⊆ 𝐴
52, 4ssexi 5328 . 2 dom (𝑥𝐴𝐵) ∈ V
6 mptexw.2 . . 3 𝐶 ∈ V
7 mptexw.3 . . . 4 𝑥𝐴 𝐵𝐶
83rnmptss 7143 . . . 4 (∀𝑥𝐴 𝐵𝐶 → ran (𝑥𝐴𝐵) ⊆ 𝐶)
97, 8ax-mp 5 . . 3 ran (𝑥𝐴𝐵) ⊆ 𝐶
106, 9ssexi 5328 . 2 ran (𝑥𝐴𝐵) ∈ V
11 funexw 7975 . 2 ((Fun (𝑥𝐴𝐵) ∧ dom (𝑥𝐴𝐵) ∈ V ∧ ran (𝑥𝐴𝐵) ∈ V) → (𝑥𝐴𝐵) ∈ V)
121, 5, 10, 11mp3an 1460 1 (𝑥𝐴𝐵) ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  wral 3059  Vcvv 3478  wss 3963  cmpt 5231  dom cdm 5689  ran crn 5690  Fun wfun 6557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565  df-fn 6566  df-f 6567
This theorem is referenced by:  grpinvfval  19009  odfval  19565
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