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Theorem resfunexgALT 6001
 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 5634 but requires ax-pow 4093 and ax-un 4350. (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 4837 . . . 4 (𝐵𝐶 → dom (𝐴𝐵) ∈ V)
21adantl 275 . . 3 ((Fun 𝐴𝐵𝐶) → dom (𝐴𝐵) ∈ V)
3 df-ima 4547 . . . 4 (𝐴𝐵) = ran (𝐴𝐵)
4 funimaexg 5202 . . . 4 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
53, 4eqeltrrid 2225 . . 3 ((Fun 𝐴𝐵𝐶) → ran (𝐴𝐵) ∈ V)
62, 5jca 304 . 2 ((Fun 𝐴𝐵𝐶) → (dom (𝐴𝐵) ∈ V ∧ ran (𝐴𝐵) ∈ V))
7 xpexg 4648 . 2 ((dom (𝐴𝐵) ∈ V ∧ ran (𝐴𝐵) ∈ V) → (dom (𝐴𝐵) × ran (𝐴𝐵)) ∈ V)
8 relres 4842 . . . 4 Rel (𝐴𝐵)
9 relssdmrn 5054 . . . 4 (Rel (𝐴𝐵) → (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)))
108, 9ax-mp 5 . . 3 (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵))
11 ssexg 4062 . . 3 (((𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)) ∧ (dom (𝐴𝐵) × ran (𝐴𝐵)) ∈ V) → (𝐴𝐵) ∈ V)
1210, 11mpan 420 . 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 1480  Vcvv 2681   ⊆ wss 3066   × cxp 4532  dom cdm 4534  ran crn 4535   ↾ cres 4536   “ cima 4537  Rel wrel 4539  Fun wfun 5112 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-fun 5120 This theorem is referenced by: (None)
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