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Theorem nff 5221
 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:A–→B

Proof of Theorem nff
StepHypRef Expression
1 df-f 4791 . 2 (F:A–→B ↔ (F Fn A ran F B))
2 nff.1 . . . 4 xF
3 nff.2 . . . 4 xA
42, 3nffn 5180 . . 3 x F Fn A
52nfrn 4963 . . . 4 xran F
6 nff.3 . . . 4 xB
75, 6nfss 3266 . . 3 xran F B
84, 7nfan 1824 . 2 x(F Fn A ran F B)
91, 8nfxfr 1570 1 x F:A–→B
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358  Ⅎwnf 1544  Ⅎwnfc 2476   ⊆ wss 3257  ran crn 4773   Fn wfn 4776  –→wf 4777 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-f 4791 This theorem is referenced by:  nff1  5256
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