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Theorem resfunexg 5303
 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 4863 . . . . 5 (Fun A → Fun (AB))
2 funfvex 5113 . . . . . 6 ((Fun (AB) x dom (AB)) → ((AB)‘x) V)
32ralrimiva 2366 . . . . 5 (Fun (AB) → x dom (AB)((AB)‘x) V)
4 fnasrng 5264 . . . . 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 4853 . . . . 5 (Fun (AB) ↔ (AB) Fn dom (AB))
97, 8sylib 127 . . . 4 ((Fun A B 𝐶) → (AB) Fn dom (AB))
10 dffn5im 5140 . . . 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 4601 . . . . 5 ((x dom (AB) ↦ ⟨x, ((AB)‘x)⟩) “ dom (x dom (AB) ↦ ⟨x, ((AB)‘x)⟩)) = ran (x dom (AB) ↦ ⟨x, ((AB)‘x)⟩)
13 vex 2534 . . . . . . . . 9 x V
14 opexgOLD 3935 . . . . . . . . 9 ((x V ((AB)‘x) V) → ⟨x, ((AB)‘x)⟩ V)
1513, 2, 14sylancr 395 . . . . . . . 8 ((Fun (AB) x dom (AB)) → ⟨x, ((AB)‘x)⟩ V)
1615ralrimiva 2366 . . . . . . 7 (Fun (AB) → x dom (AB)⟨x, ((AB)‘x)⟩ V)
17 dmmptg 4741 . . . . . . 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 4591 . . . . 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 2065 . . . 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 2060 . 2 ((Fun A B 𝐶) → (AB) = ((x dom (AB) ↦ ⟨x, ((AB)‘x)⟩) “ dom (AB)))
23 funmpt 4860 . . 3 Fun (x dom (AB) ↦ ⟨x, ((AB)‘x)⟩)
24 dmresexg 4557 . . . 4 (B 𝐶 → dom (AB) V)
2524adantl 262 . . 3 ((Fun A B 𝐶) → dom (AB) V)
26 funimaexg 4905 . . 3 ((Fun (x dom (AB) ↦ ⟨x, ((AB)‘x)⟩) dom (AB) V) → ((x dom (AB) ↦ ⟨x, ((AB)‘x)⟩) “ dom (AB)) V)
2723, 25, 26sylancr 395 . 2 ((Fun A B 𝐶) → ((x dom (AB) ↦ ⟨x, ((AB)‘x)⟩) “ dom (AB)) V)
2822, 27eqeltrd 2092 1 ((Fun A B 𝐶) → (AB) V)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1226   ∈ wcel 1370  ∀wral 2280  Vcvv 2531  ⟨cop 3349   ↦ cmpt 3788  dom cdm 4268  ran crn 4269   ↾ cres 4270   “ cima 4271  Fun wfun 4819   Fn wfn 4820  ‘cfv 4825 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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-coll 3842  ax-sep 3845  ax-pow 3897  ax-pr 3914 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-reu 2287  df-rab 2289  df-v 2533  df-sbc 2738  df-csb 2826  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-iun 3629  df-br 3735  df-opab 3789  df-mpt 3790  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832  df-fv 4833 This theorem is referenced by:  fnex  5304  ofexg  5635  cofunexg  5657  rdgivallem  5884  frecsuclem3  5902
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