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Theorem fnexALT 5682
Description: If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of funimaexg 4926. This version of fnex 5326 uses ax-pow 3918 and ax-un 4136, whereas fnex 5326 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
fnexALT ((𝐹 Fn A A B) → 𝐹 V)

Proof of Theorem fnexALT
StepHypRef Expression
1 fnrel 4940 . . . 4 (𝐹 Fn A → Rel 𝐹)
2 relssdmrn 4784 . . . 4 (Rel 𝐹𝐹 ⊆ (dom 𝐹 × ran 𝐹))
31, 2syl 14 . . 3 (𝐹 Fn A𝐹 ⊆ (dom 𝐹 × ran 𝐹))
43adantr 261 . 2 ((𝐹 Fn A A B) → 𝐹 ⊆ (dom 𝐹 × ran 𝐹))
5 fndm 4941 . . . . 5 (𝐹 Fn A → dom 𝐹 = A)
65eleq1d 2103 . . . 4 (𝐹 Fn A → (dom 𝐹 BA B))
76biimpar 281 . . 3 ((𝐹 Fn A A B) → dom 𝐹 B)
8 fnfun 4939 . . . . 5 (𝐹 Fn A → Fun 𝐹)
9 funimaexg 4926 . . . . 5 ((Fun 𝐹 A B) → (𝐹A) V)
108, 9sylan 267 . . . 4 ((𝐹 Fn A A B) → (𝐹A) V)
11 imadmrn 4621 . . . . . . 7 (𝐹 “ dom 𝐹) = ran 𝐹
125imaeq2d 4611 . . . . . . 7 (𝐹 Fn A → (𝐹 “ dom 𝐹) = (𝐹A))
1311, 12syl5eqr 2083 . . . . . 6 (𝐹 Fn A → ran 𝐹 = (𝐹A))
1413eleq1d 2103 . . . . 5 (𝐹 Fn A → (ran 𝐹 V ↔ (𝐹A) V))
1514biimpar 281 . . . 4 ((𝐹 Fn A (𝐹A) V) → ran 𝐹 V)
1610, 15syldan 266 . . 3 ((𝐹 Fn A A B) → ran 𝐹 V)
17 xpexg 4395 . . 3 ((dom 𝐹 B ran 𝐹 V) → (dom 𝐹 × ran 𝐹) V)
187, 16, 17syl2anc 391 . 2 ((𝐹 Fn A A B) → (dom 𝐹 × ran 𝐹) V)
19 ssexg 3887 . 2 ((𝐹 ⊆ (dom 𝐹 × ran 𝐹) (dom 𝐹 × ran 𝐹) V) → 𝐹 V)
204, 18, 19syl2anc 391 1 ((𝐹 Fn A A B) → 𝐹 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  cima 4291  Rel wrel 4293  Fun wfun 4839   Fn wfn 4840
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  df-fn 4848
This theorem is referenced by: (None)
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