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Theorem fvelrn 5219
Description: A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.)
Assertion
Ref Expression
fvelrn ((Fun 𝐹 A dom 𝐹) → (𝐹A) ran 𝐹)

Proof of Theorem fvelrn
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2078 . . . . 5 (x = A → (x dom 𝐹A dom 𝐹))
21anbi2d 440 . . . 4 (x = A → ((Fun 𝐹 x dom 𝐹) ↔ (Fun 𝐹 A dom 𝐹)))
3 fveq2 5099 . . . . 5 (x = A → (𝐹x) = (𝐹A))
43eleq1d 2084 . . . 4 (x = A → ((𝐹x) ran 𝐹 ↔ (𝐹A) ran 𝐹))
52, 4imbi12d 223 . . 3 (x = A → (((Fun 𝐹 x dom 𝐹) → (𝐹x) ran 𝐹) ↔ ((Fun 𝐹 A dom 𝐹) → (𝐹A) ran 𝐹)))
6 funfvop 5200 . . . . 5 ((Fun 𝐹 x dom 𝐹) → ⟨x, (𝐹x)⟩ 𝐹)
7 vex 2534 . . . . . 6 x V
8 opeq1 3519 . . . . . . 7 (y = x → ⟨y, (𝐹x)⟩ = ⟨x, (𝐹x)⟩)
98eleq1d 2084 . . . . . 6 (y = x → (⟨y, (𝐹x)⟩ 𝐹 ↔ ⟨x, (𝐹x)⟩ 𝐹))
107, 9spcev 2620 . . . . 5 (⟨x, (𝐹x)⟩ 𝐹yy, (𝐹x)⟩ 𝐹)
116, 10syl 14 . . . 4 ((Fun 𝐹 x dom 𝐹) → yy, (𝐹x)⟩ 𝐹)
12 funfvex 5113 . . . . 5 ((Fun 𝐹 x dom 𝐹) → (𝐹x) V)
13 elrn2g 4448 . . . . 5 ((𝐹x) V → ((𝐹x) ran 𝐹yy, (𝐹x)⟩ 𝐹))
1412, 13syl 14 . . . 4 ((Fun 𝐹 x dom 𝐹) → ((𝐹x) ran 𝐹yy, (𝐹x)⟩ 𝐹))
1511, 14mpbird 156 . . 3 ((Fun 𝐹 x dom 𝐹) → (𝐹x) ran 𝐹)
165, 15vtoclg 2586 . 2 (A dom 𝐹 → ((Fun 𝐹 A dom 𝐹) → (𝐹A) ran 𝐹))
1716anabsi7 502 1 ((Fun 𝐹 A dom 𝐹) → (𝐹A) ran 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1226  wex 1358   wcel 1370  Vcvv 2531  cop 3349  dom cdm 4268  ran crn 4269  Fun wfun 4819  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-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-v 2533  df-sbc 2738  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-br 3735  df-opab 3789  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-iota 4790  df-fun 4827  df-fn 4828  df-fv 4833
This theorem is referenced by:  fnfvelrn  5220  eldmrexrn  5229  funfvima  5311  elunirn  5326
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