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Theorem resfunexgALT 5765
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 5410 but requires ax-pow 3955 and ax-un 4198. (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 4662 . . . 4 (𝐵𝐶 → dom (𝐴𝐵) ∈ V)
21adantl 266 . . 3 ((Fun 𝐴𝐵𝐶) → dom (𝐴𝐵) ∈ V)
3 df-ima 4386 . . . 4 (𝐴𝐵) = ran (𝐴𝐵)
4 funimaexg 5011 . . . 4 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
53, 4syl5eqelr 2141 . . 3 ((Fun 𝐴𝐵𝐶) → ran (𝐴𝐵) ∈ V)
62, 5jca 294 . 2 ((Fun 𝐴𝐵𝐶) → (dom (𝐴𝐵) ∈ V ∧ ran (𝐴𝐵) ∈ V))
7 xpexg 4480 . 2 ((dom (𝐴𝐵) ∈ V ∧ ran (𝐴𝐵) ∈ V) → (dom (𝐴𝐵) × ran (𝐴𝐵)) ∈ V)
8 relres 4667 . . . 4 Rel (𝐴𝐵)
9 relssdmrn 4869 . . . 4 (Rel (𝐴𝐵) → (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)))
108, 9ax-mp 7 . . 3 (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵))
11 ssexg 3924 . . 3 (((𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)) ∧ (dom (𝐴𝐵) × ran (𝐴𝐵)) ∈ V) → (𝐴𝐵) ∈ V)
1210, 11mpan 408 . 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 101  wcel 1409  Vcvv 2574  wss 2945   × cxp 4371  dom cdm 4373  ran crn 4374  cres 4375  cima 4376  Rel wrel 4378  Fun wfun 4924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-fun 4932
This theorem is referenced by: (None)
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