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Theorem resfunexgALT 5679
 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 5325 but requires ax-pow 3918 and ax-un 4136. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
resfunexgALT ((Fun A B 𝐶) → (AB) V)

Proof of Theorem resfunexgALT
StepHypRef Expression
1 dmresexg 4577 . . . 4 (B 𝐶 → dom (AB) V)
21adantl 262 . . 3 ((Fun A B 𝐶) → dom (AB) V)
3 df-ima 4301 . . . 4 (AB) = ran (AB)
4 funimaexg 4926 . . . 4 ((Fun A B 𝐶) → (AB) V)
53, 4syl5eqelr 2122 . . 3 ((Fun A B 𝐶) → ran (AB) V)
62, 5jca 290 . 2 ((Fun A B 𝐶) → (dom (AB) V ran (AB) V))
7 xpexg 4395 . 2 ((dom (AB) V ran (AB) V) → (dom (AB) × ran (AB)) V)
8 relres 4582 . . . 4 Rel (AB)
9 relssdmrn 4784 . . . 4 (Rel (AB) → (AB) ⊆ (dom (AB) × ran (AB)))
108, 9ax-mp 7 . . 3 (AB) ⊆ (dom (AB) × ran (AB))
11 ssexg 3887 . . 3 (((AB) ⊆ (dom (AB) × ran (AB)) (dom (AB) × ran (AB)) V) → (AB) V)
1210, 11mpan 400 . 2 ((dom (AB) × ran (AB)) V → (AB) V)
136, 7, 123syl 17 1 ((Fun A B 𝐶) → (AB) V)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1390  Vcvv 2551   ⊆ wss 2911   × cxp 4286  dom cdm 4288  ran crn 4289   ↾ cres 4290   “ cima 4291  Rel wrel 4293  Fun wfun 4839 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-13 1401  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  ax-un 4136 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-v 2553  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-br 3756  df-opab 3810  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-fun 4847 This theorem is referenced by: (None)
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