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Theorem ffnfv 5266
 Description: A function maps to a class to which all values belong. (Contributed by NM, 3-Dec-2003.)
Assertion
Ref Expression
ffnfv (𝐹:AB ↔ (𝐹 Fn A x A (𝐹x) B))
Distinct variable groups:   x,A   x,B   x,𝐹

Proof of Theorem ffnfv
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 ffn 4989 . . 3 (𝐹:AB𝐹 Fn A)
2 ffvelrn 5243 . . . 4 ((𝐹:AB x A) → (𝐹x) B)
32ralrimiva 2386 . . 3 (𝐹:ABx A (𝐹x) B)
41, 3jca 290 . 2 (𝐹:AB → (𝐹 Fn A x A (𝐹x) B))
5 simpl 102 . . 3 ((𝐹 Fn A x A (𝐹x) B) → 𝐹 Fn A)
6 fvelrnb 5164 . . . . . 6 (𝐹 Fn A → (y ran 𝐹x A (𝐹x) = y))
76biimpd 132 . . . . 5 (𝐹 Fn A → (y ran 𝐹x A (𝐹x) = y))
8 nfra1 2349 . . . . . 6 xx A (𝐹x) B
9 nfv 1418 . . . . . 6 x y B
10 rsp 2363 . . . . . . 7 (x A (𝐹x) B → (x A → (𝐹x) B))
11 eleq1 2097 . . . . . . . 8 ((𝐹x) = y → ((𝐹x) By B))
1211biimpcd 148 . . . . . . 7 ((𝐹x) B → ((𝐹x) = yy B))
1310, 12syl6 29 . . . . . 6 (x A (𝐹x) B → (x A → ((𝐹x) = yy B)))
148, 9, 13rexlimd 2424 . . . . 5 (x A (𝐹x) B → (x A (𝐹x) = yy B))
157, 14sylan9 389 . . . 4 ((𝐹 Fn A x A (𝐹x) B) → (y ran 𝐹y B))
1615ssrdv 2945 . . 3 ((𝐹 Fn A x A (𝐹x) B) → ran 𝐹B)
17 df-f 4849 . . 3 (𝐹:AB ↔ (𝐹 Fn A ran 𝐹B))
185, 16, 17sylanbrc 394 . 2 ((𝐹 Fn A x A (𝐹x) B) → 𝐹:AB)
194, 18impbii 117 1 (𝐹:AB ↔ (𝐹 Fn A x A (𝐹x) B))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∀wral 2300  ∃wrex 2301   ⊆ wss 2911  ran crn 4289   Fn wfn 4840  ⟶wf 4841  ‘cfv 4845 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-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 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-sbc 2759  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-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853 This theorem is referenced by:  ffnfvf  5267  fnfvrnss  5268  fmpt2d  5270  ffnov  5547  cnref1o  8357
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