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Theorem ffnfv 5571
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 5267 . . 3 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 ffvelrn 5546 . . . 4 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
32ralrimiva 2503 . . 3 (𝐹:𝐴𝐵 → ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)
41, 3jca 304 . 2 (𝐹:𝐴𝐵 → (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
5 simpl 108 . . 3 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → 𝐹 Fn 𝐴)
6 fvelrnb 5462 . . . . . 6 (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝑦))
76biimpd 143 . . . . 5 (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 → ∃𝑥𝐴 (𝐹𝑥) = 𝑦))
8 nfra1 2464 . . . . . 6 𝑥𝑥𝐴 (𝐹𝑥) ∈ 𝐵
9 nfv 1508 . . . . . 6 𝑥 𝑦𝐵
10 rsp 2478 . . . . . . 7 (∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵 → (𝑥𝐴 → (𝐹𝑥) ∈ 𝐵))
11 eleq1 2200 . . . . . . . 8 ((𝐹𝑥) = 𝑦 → ((𝐹𝑥) ∈ 𝐵𝑦𝐵))
1211biimpcd 158 . . . . . . 7 ((𝐹𝑥) ∈ 𝐵 → ((𝐹𝑥) = 𝑦𝑦𝐵))
1310, 12syl6 33 . . . . . 6 (∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵 → (𝑥𝐴 → ((𝐹𝑥) = 𝑦𝑦𝐵)))
148, 9, 13rexlimd 2544 . . . . 5 (∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵 → (∃𝑥𝐴 (𝐹𝑥) = 𝑦𝑦𝐵))
157, 14sylan9 406 . . . 4 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → (𝑦 ∈ ran 𝐹𝑦𝐵))
1615ssrdv 3098 . . 3 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → ran 𝐹𝐵)
17 df-f 5122 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
185, 16, 17sylanbrc 413 . 2 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → 𝐹:𝐴𝐵)
194, 18impbii 125 1 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wcel 1480  wral 2414  wrex 2415  wss 3066  ran crn 4535   Fn wfn 5113  wf 5114  cfv 5118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126
This theorem is referenced by:  ffnfvf  5572  fnfvrnss  5573  fmpt2d  5575  ffnov  5868  elixpconst  6593  elixpsn  6622  ctssdccl  6989  cnref1o  9433  shftf  10595  eff2  11375  reeff1  11396  dvfre  12832  isomninnlem  13214
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