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Theorem mptexw 7886
Description: Weak version of mptex 7174 that holds without ax-rep 5243. 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 6540 . 2 Fun (𝑥𝐴𝐵)
2 mptexw.1 . . 3 𝐴 ∈ V
3 eqid 2733 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43dmmptss 6194 . . 3 dom (𝑥𝐴𝐵) ⊆ 𝐴
52, 4ssexi 5280 . 2 dom (𝑥𝐴𝐵) ∈ V
6 mptexw.2 . . 3 𝐶 ∈ V
7 mptexw.3 . . . 4 𝑥𝐴 𝐵𝐶
83rnmptss 7071 . . . 4 (∀𝑥𝐴 𝐵𝐶 → ran (𝑥𝐴𝐵) ⊆ 𝐶)
97, 8ax-mp 5 . . 3 ran (𝑥𝐴𝐵) ⊆ 𝐶
106, 9ssexi 5280 . 2 ran (𝑥𝐴𝐵) ∈ V
11 funexw 7885 . 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 3061  Vcvv 3444  wss 3911  cmpt 5189  dom cdm 5634  ran crn 5635  Fun wfun 6491
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 2704  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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  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-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-fun 6499  df-fn 6500  df-f 6501
This theorem is referenced by:  grpinvfval  18794  odfval  19319
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