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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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