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Theorem fnexALT 6109
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 5299. This version of fnex 5737 uses ax-pow 4173 and ax-un 4432, whereas fnex 5737 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 5313 . . . 4 (𝐹 Fn 𝐴 → Rel 𝐹)
2 relssdmrn 5148 . . . 4 (Rel 𝐹𝐹 ⊆ (dom 𝐹 × ran 𝐹))
31, 2syl 14 . . 3 (𝐹 Fn 𝐴𝐹 ⊆ (dom 𝐹 × ran 𝐹))
43adantr 276 . 2 ((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ⊆ (dom 𝐹 × ran 𝐹))
5 fndm 5314 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
65eleq1d 2246 . . . 4 (𝐹 Fn 𝐴 → (dom 𝐹𝐵𝐴𝐵))
76biimpar 297 . . 3 ((𝐹 Fn 𝐴𝐴𝐵) → dom 𝐹𝐵)
8 fnfun 5312 . . . . 5 (𝐹 Fn 𝐴 → Fun 𝐹)
9 funimaexg 5299 . . . . 5 ((Fun 𝐹𝐴𝐵) → (𝐹𝐴) ∈ V)
108, 9sylan 283 . . . 4 ((𝐹 Fn 𝐴𝐴𝐵) → (𝐹𝐴) ∈ V)
11 imadmrn 4979 . . . . . . 7 (𝐹 “ dom 𝐹) = ran 𝐹
125imaeq2d 4969 . . . . . . 7 (𝐹 Fn 𝐴 → (𝐹 “ dom 𝐹) = (𝐹𝐴))
1311, 12eqtr3id 2224 . . . . . 6 (𝐹 Fn 𝐴 → ran 𝐹 = (𝐹𝐴))
1413eleq1d 2246 . . . . 5 (𝐹 Fn 𝐴 → (ran 𝐹 ∈ V ↔ (𝐹𝐴) ∈ V))
1514biimpar 297 . . . 4 ((𝐹 Fn 𝐴 ∧ (𝐹𝐴) ∈ V) → ran 𝐹 ∈ V)
1610, 15syldan 282 . . 3 ((𝐹 Fn 𝐴𝐴𝐵) → ran 𝐹 ∈ V)
17 xpexg 4739 . . 3 ((dom 𝐹𝐵 ∧ ran 𝐹 ∈ V) → (dom 𝐹 × ran 𝐹) ∈ V)
187, 16, 17syl2anc 411 . 2 ((𝐹 Fn 𝐴𝐴𝐵) → (dom 𝐹 × ran 𝐹) ∈ V)
19 ssexg 4141 . 2 ((𝐹 ⊆ (dom 𝐹 × ran 𝐹) ∧ (dom 𝐹 × ran 𝐹) ∈ V) → 𝐹 ∈ V)
204, 18, 19syl2anc 411 1 ((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wcel 2148  Vcvv 2737  wss 3129   × cxp 4623  dom cdm 4625  ran crn 4626  cima 4628  Rel wrel 4630  Fun wfun 5209   Fn wfn 5210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-fun 5217  df-fn 5218
This theorem is referenced by: (None)
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