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Theorem fnexALT 5979
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 5177. This version of fnex 5610 uses ax-pow 4068 and ax-un 4325, whereas fnex 5610 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
fnexALT ((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ∈ V)

Proof of Theorem fnexALT
StepHypRef Expression
1 fnrel 5191 . . . 4 (𝐹 Fn 𝐴 → Rel 𝐹)
2 relssdmrn 5029 . . . 4 (Rel 𝐹𝐹 ⊆ (dom 𝐹 × ran 𝐹))
31, 2syl 14 . . 3 (𝐹 Fn 𝐴𝐹 ⊆ (dom 𝐹 × ran 𝐹))
43adantr 274 . 2 ((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ⊆ (dom 𝐹 × ran 𝐹))
5 fndm 5192 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
65eleq1d 2186 . . . 4 (𝐹 Fn 𝐴 → (dom 𝐹𝐵𝐴𝐵))
76biimpar 295 . . 3 ((𝐹 Fn 𝐴𝐴𝐵) → dom 𝐹𝐵)
8 fnfun 5190 . . . . 5 (𝐹 Fn 𝐴 → Fun 𝐹)
9 funimaexg 5177 . . . . 5 ((Fun 𝐹𝐴𝐵) → (𝐹𝐴) ∈ V)
108, 9sylan 281 . . . 4 ((𝐹 Fn 𝐴𝐴𝐵) → (𝐹𝐴) ∈ V)
11 imadmrn 4861 . . . . . . 7 (𝐹 “ dom 𝐹) = ran 𝐹
125imaeq2d 4851 . . . . . . 7 (𝐹 Fn 𝐴 → (𝐹 “ dom 𝐹) = (𝐹𝐴))
1311, 12syl5eqr 2164 . . . . . 6 (𝐹 Fn 𝐴 → ran 𝐹 = (𝐹𝐴))
1413eleq1d 2186 . . . . 5 (𝐹 Fn 𝐴 → (ran 𝐹 ∈ V ↔ (𝐹𝐴) ∈ V))
1514biimpar 295 . . . 4 ((𝐹 Fn 𝐴 ∧ (𝐹𝐴) ∈ V) → ran 𝐹 ∈ V)
1610, 15syldan 280 . . 3 ((𝐹 Fn 𝐴𝐴𝐵) → ran 𝐹 ∈ V)
17 xpexg 4623 . . 3 ((dom 𝐹𝐵 ∧ ran 𝐹 ∈ V) → (dom 𝐹 × ran 𝐹) ∈ V)
187, 16, 17syl2anc 408 . 2 ((𝐹 Fn 𝐴𝐴𝐵) → (dom 𝐹 × ran 𝐹) ∈ V)
19 ssexg 4037 . 2 ((𝐹 ⊆ (dom 𝐹 × ran 𝐹) ∧ (dom 𝐹 × ran 𝐹) ∈ V) → 𝐹 ∈ V)
204, 18, 19syl2anc 408 1 ((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 1465  Vcvv 2660  wss 3041   × cxp 4507  dom cdm 4509  ran crn 4510  cima 4512  Rel wrel 4514  Fun wfun 5087   Fn wfn 5088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-fun 5095  df-fn 5096
This theorem is referenced by: (None)
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