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Theorem nff 4984
 Description: Bound-variable hypothesis builder for a mapping. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
nff.1 x𝐹
nff.2 xA
nff.3 xB
Assertion
Ref Expression
nff x 𝐹:AB

Proof of Theorem nff
StepHypRef Expression
1 df-f 4848 . 2 (𝐹:AB ↔ (𝐹 Fn A ran 𝐹B))
2 nff.1 . . . 4 x𝐹
3 nff.2 . . . 4 xA
42, 3nffn 4936 . . 3 x 𝐹 Fn A
52nfrn 4521 . . . 4 xran 𝐹
6 nff.3 . . . 4 xB
75, 6nfss 2932 . . 3 xran 𝐹B
84, 7nfan 1454 . 2 x(𝐹 Fn A ran 𝐹B)
91, 8nfxfr 1360 1 x 𝐹:AB
 Colors of variables: wff set class Syntax hints:   ∧ wa 97  Ⅎwnf 1346  Ⅎwnfc 2162   ⊆ wss 2911  ran crn 4288   Fn wfn 4839  ⟶wf 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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3372  df-pr 3373  df-op 3375  df-br 3755  df-opab 3809  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-rn 4298  df-fun 4846  df-fn 4847  df-f 4848 This theorem is referenced by:  nff1  5031
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