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Theorem ffnfv 5675
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 ∧ ∀𝑥𝐴 (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 5366 . . 3 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 ffvelcdm 5650 . . . 4 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
32ralrimiva 2550 . . 3 (𝐹:𝐴𝐵 → ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)
41, 3jca 306 . 2 (𝐹:𝐴𝐵 → (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
5 simpl 109 . . 3 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → 𝐹 Fn 𝐴)
6 fvelrnb 5564 . . . . . 6 (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝑦))
76biimpd 144 . . . . 5 (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 → ∃𝑥𝐴 (𝐹𝑥) = 𝑦))
8 nfra1 2508 . . . . . 6 𝑥𝑥𝐴 (𝐹𝑥) ∈ 𝐵
9 nfv 1528 . . . . . 6 𝑥 𝑦𝐵
10 rsp 2524 . . . . . . 7 (∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵 → (𝑥𝐴 → (𝐹𝑥) ∈ 𝐵))
11 eleq1 2240 . . . . . . . 8 ((𝐹𝑥) = 𝑦 → ((𝐹𝑥) ∈ 𝐵𝑦𝐵))
1211biimpcd 159 . . . . . . 7 ((𝐹𝑥) ∈ 𝐵 → ((𝐹𝑥) = 𝑦𝑦𝐵))
1310, 12syl6 33 . . . . . 6 (∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵 → (𝑥𝐴 → ((𝐹𝑥) = 𝑦𝑦𝐵)))
148, 9, 13rexlimd 2591 . . . . 5 (∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵 → (∃𝑥𝐴 (𝐹𝑥) = 𝑦𝑦𝐵))
157, 14sylan9 409 . . . 4 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → (𝑦 ∈ ran 𝐹𝑦𝐵))
1615ssrdv 3162 . . 3 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → ran 𝐹𝐵)
17 df-f 5221 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
185, 16, 17sylanbrc 417 . 2 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → 𝐹:𝐴𝐵)
194, 18impbii 126 1 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wral 2455  wrex 2456  wss 3130  ran crn 4628   Fn wfn 5212  wf 5213  cfv 5217
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-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210
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 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225
This theorem is referenced by:  ffnfvf  5676  fnfvrnss  5677  fmpt2d  5679  ffnov  5979  elixpconst  6706  elixpsn  6735  ctssdccl  7110  cnref1o  9650  shftf  10839  eff2  11688  reeff1  11708  1arith  12365  ptex  12713  xpscf  12766  mgpf  13194  dvfre  14177  ioocosf1o  14278  012of  14748  2o01f  14749
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