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Theorem resfunexg 5325
Description: The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. (Contributed by NM, 7-Apr-1995.) (Revised by Mario Carneiro, 22-Jun-2013.)
Assertion
Ref Expression
resfunexg ((Fun A B 𝐶) → (AB) V)

Proof of Theorem resfunexg
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 funres 4884 . . . . 5 (Fun A → Fun (AB))
2 funfvex 5135 . . . . . 6 ((Fun (AB) x dom (AB)) → ((AB)‘x) V)
32ralrimiva 2386 . . . . 5 (Fun (AB) → x dom (AB)((AB)‘x) V)
4 fnasrng 5286 . . . . 5 (x dom (AB)((AB)‘x) V → (x dom (AB) ↦ ((AB)‘x)) = ran (x dom (AB) ↦ ⟨x, ((AB)‘x)⟩))
51, 3, 43syl 17 . . . 4 (Fun A → (x dom (AB) ↦ ((AB)‘x)) = ran (x dom (AB) ↦ ⟨x, ((AB)‘x)⟩))
65adantr 261 . . 3 ((Fun A B 𝐶) → (x dom (AB) ↦ ((AB)‘x)) = ran (x dom (AB) ↦ ⟨x, ((AB)‘x)⟩))
71adantr 261 . . . . 5 ((Fun A B 𝐶) → Fun (AB))
8 funfn 4874 . . . . 5 (Fun (AB) ↔ (AB) Fn dom (AB))
97, 8sylib 127 . . . 4 ((Fun A B 𝐶) → (AB) Fn dom (AB))
10 dffn5im 5162 . . . 4 ((AB) Fn dom (AB) → (AB) = (x dom (AB) ↦ ((AB)‘x)))
119, 10syl 14 . . 3 ((Fun A B 𝐶) → (AB) = (x dom (AB) ↦ ((AB)‘x)))
12 imadmrn 4621 . . . . 5 ((x dom (AB) ↦ ⟨x, ((AB)‘x)⟩) “ dom (x dom (AB) ↦ ⟨x, ((AB)‘x)⟩)) = ran (x dom (AB) ↦ ⟨x, ((AB)‘x)⟩)
13 vex 2554 . . . . . . . . 9 x V
14 opexgOLD 3956 . . . . . . . . 9 ((x V ((AB)‘x) V) → ⟨x, ((AB)‘x)⟩ V)
1513, 2, 14sylancr 393 . . . . . . . 8 ((Fun (AB) x dom (AB)) → ⟨x, ((AB)‘x)⟩ V)
1615ralrimiva 2386 . . . . . . 7 (Fun (AB) → x dom (AB)⟨x, ((AB)‘x)⟩ V)
17 dmmptg 4761 . . . . . . 7 (x dom (AB)⟨x, ((AB)‘x)⟩ V → dom (x dom (AB) ↦ ⟨x, ((AB)‘x)⟩) = dom (AB))
181, 16, 173syl 17 . . . . . 6 (Fun A → dom (x dom (AB) ↦ ⟨x, ((AB)‘x)⟩) = dom (AB))
1918imaeq2d 4611 . . . . 5 (Fun A → ((x dom (AB) ↦ ⟨x, ((AB)‘x)⟩) “ dom (x dom (AB) ↦ ⟨x, ((AB)‘x)⟩)) = ((x dom (AB) ↦ ⟨x, ((AB)‘x)⟩) “ dom (AB)))
2012, 19syl5reqr 2084 . . . 4 (Fun A → ((x dom (AB) ↦ ⟨x, ((AB)‘x)⟩) “ dom (AB)) = ran (x dom (AB) ↦ ⟨x, ((AB)‘x)⟩))
2120adantr 261 . . 3 ((Fun A B 𝐶) → ((x dom (AB) ↦ ⟨x, ((AB)‘x)⟩) “ dom (AB)) = ran (x dom (AB) ↦ ⟨x, ((AB)‘x)⟩))
226, 11, 213eqtr4d 2079 . 2 ((Fun A B 𝐶) → (AB) = ((x dom (AB) ↦ ⟨x, ((AB)‘x)⟩) “ dom (AB)))
23 funmpt 4881 . . 3 Fun (x dom (AB) ↦ ⟨x, ((AB)‘x)⟩)
24 dmresexg 4577 . . . 4 (B 𝐶 → dom (AB) V)
2524adantl 262 . . 3 ((Fun A B 𝐶) → dom (AB) V)
26 funimaexg 4926 . . 3 ((Fun (x dom (AB) ↦ ⟨x, ((AB)‘x)⟩) dom (AB) V) → ((x dom (AB) ↦ ⟨x, ((AB)‘x)⟩) “ dom (AB)) V)
2723, 25, 26sylancr 393 . 2 ((Fun A B 𝐶) → ((x dom (AB) ↦ ⟨x, ((AB)‘x)⟩) “ dom (AB)) V)
2822, 27eqeltrd 2111 1 ((Fun A B 𝐶) → (AB) V)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  wral 2300  Vcvv 2551  cop 3370  cmpt 3809  dom cdm 4288  ran crn 4289  cres 4290  cima 4291  Fun wfun 4839   Fn wfn 4840  cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853
This theorem is referenced by:  fnex  5326  ofexg  5658  cofunexg  5680  rdgivallem  5908  frecsuclem3  5929
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