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Theorem mptexw 6127
Description: Weak version of mptex 5755 that holds without ax-coll 4130. 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 5266 . 2 Fun (𝑥𝐴𝐵)
2 mptexw.1 . . 3 𝐴 ∈ V
3 eqid 2187 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43dmmptss 5137 . . 3 dom (𝑥𝐴𝐵) ⊆ 𝐴
52, 4ssexi 4153 . 2 dom (𝑥𝐴𝐵) ∈ V
6 mptexw.2 . . 3 𝐶 ∈ V
7 mptexw.3 . . . 4 𝑥𝐴 𝐵𝐶
83rnmptss 5690 . . . 4 (∀𝑥𝐴 𝐵𝐶 → ran (𝑥𝐴𝐵) ⊆ 𝐶)
97, 8ax-mp 5 . . 3 ran (𝑥𝐴𝐵) ⊆ 𝐶
106, 9ssexi 4153 . 2 ran (𝑥𝐴𝐵) ∈ V
11 funexw 6126 . 2 ((Fun (𝑥𝐴𝐵) ∧ dom (𝑥𝐴𝐵) ∈ V ∧ ran (𝑥𝐴𝐵) ∈ V) → (𝑥𝐴𝐵) ∈ V)
121, 5, 10, 11mp3an 1347 1 (𝑥𝐴𝐵) ∈ V
Colors of variables: wff set class
Syntax hints:  wcel 2158  wral 2465  Vcvv 2749  wss 3141  cmpt 4076  dom cdm 4638  ran crn 4639  Fun wfun 5222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fv 5236
This theorem is referenced by: (None)
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