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Theorem mptexw 7932
Description: Weak version of mptex 7216 that holds without ax-rep 5275. 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 6576 . 2 Fun (𝑥𝐴𝐵)
2 mptexw.1 . . 3 𝐴 ∈ V
3 eqid 2724 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43dmmptss 6230 . . 3 dom (𝑥𝐴𝐵) ⊆ 𝐴
52, 4ssexi 5312 . 2 dom (𝑥𝐴𝐵) ∈ V
6 mptexw.2 . . 3 𝐶 ∈ V
7 mptexw.3 . . . 4 𝑥𝐴 𝐵𝐶
83rnmptss 7114 . . . 4 (∀𝑥𝐴 𝐵𝐶 → ran (𝑥𝐴𝐵) ⊆ 𝐶)
97, 8ax-mp 5 . . 3 ran (𝑥𝐴𝐵) ⊆ 𝐶
106, 9ssexi 5312 . 2 ran (𝑥𝐴𝐵) ∈ V
11 funexw 7931 . 2 ((Fun (𝑥𝐴𝐵) ∧ dom (𝑥𝐴𝐵) ∈ V ∧ ran (𝑥𝐴𝐵) ∈ V) → (𝑥𝐴𝐵) ∈ V)
121, 5, 10, 11mp3an 1457 1 (𝑥𝐴𝐵) ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2098  wral 3053  Vcvv 3466  wss 3940  cmpt 5221  dom cdm 5666  ran crn 5667  Fun wfun 6527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-fun 6535  df-fn 6536  df-f 6537
This theorem is referenced by:  grpinvfval  18895  odfval  19437
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