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Theorem resfunexgALT 6071
Description: The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. This version has a shorter proof than resfunexg 5701 but requires ax-pow 4148 and ax-un 4406. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
resfunexgALT ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)

Proof of Theorem resfunexgALT
StepHypRef Expression
1 dmresexg 4902 . . . 4 (𝐵𝐶 → dom (𝐴𝐵) ∈ V)
21adantl 275 . . 3 ((Fun 𝐴𝐵𝐶) → dom (𝐴𝐵) ∈ V)
3 df-ima 4612 . . . 4 (𝐴𝐵) = ran (𝐴𝐵)
4 funimaexg 5267 . . . 4 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
53, 4eqeltrrid 2252 . . 3 ((Fun 𝐴𝐵𝐶) → ran (𝐴𝐵) ∈ V)
62, 5jca 304 . 2 ((Fun 𝐴𝐵𝐶) → (dom (𝐴𝐵) ∈ V ∧ ran (𝐴𝐵) ∈ V))
7 xpexg 4713 . 2 ((dom (𝐴𝐵) ∈ V ∧ ran (𝐴𝐵) ∈ V) → (dom (𝐴𝐵) × ran (𝐴𝐵)) ∈ V)
8 relres 4907 . . . 4 Rel (𝐴𝐵)
9 relssdmrn 5119 . . . 4 (Rel (𝐴𝐵) → (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)))
108, 9ax-mp 5 . . 3 (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵))
11 ssexg 4116 . . 3 (((𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)) ∧ (dom (𝐴𝐵) × ran (𝐴𝐵)) ∈ V) → (𝐴𝐵) ∈ V)
1210, 11mpan 421 . 2 ((dom (𝐴𝐵) × ran (𝐴𝐵)) ∈ V → (𝐴𝐵) ∈ V)
136, 7, 123syl 17 1 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 2135  Vcvv 2722  wss 3112   × cxp 4597  dom cdm 4599  ran crn 4600  cres 4601  cima 4602  Rel wrel 4604  Fun wfun 5177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4092  ax-sep 4095  ax-pow 4148  ax-pr 4182  ax-un 4406
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2724  df-un 3116  df-in 3118  df-ss 3125  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-uni 3785  df-br 3978  df-opab 4039  df-id 4266  df-xp 4605  df-rel 4606  df-cnv 4607  df-co 4608  df-dm 4609  df-rn 4610  df-res 4611  df-ima 4612  df-fun 5185
This theorem is referenced by: (None)
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