ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ffnfv GIF version

Theorem ffnfv 5350
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 5073 . . 3 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 ffvelrn 5327 . . . 4 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
32ralrimiva 2409 . . 3 (𝐹:𝐴𝐵 → ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)
41, 3jca 294 . 2 (𝐹:𝐴𝐵 → (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
5 simpl 106 . . 3 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → 𝐹 Fn 𝐴)
6 fvelrnb 5248 . . . . . 6 (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝑦))
76biimpd 136 . . . . 5 (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 → ∃𝑥𝐴 (𝐹𝑥) = 𝑦))
8 nfra1 2372 . . . . . 6 𝑥𝑥𝐴 (𝐹𝑥) ∈ 𝐵
9 nfv 1437 . . . . . 6 𝑥 𝑦𝐵
10 rsp 2386 . . . . . . 7 (∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵 → (𝑥𝐴 → (𝐹𝑥) ∈ 𝐵))
11 eleq1 2116 . . . . . . . 8 ((𝐹𝑥) = 𝑦 → ((𝐹𝑥) ∈ 𝐵𝑦𝐵))
1211biimpcd 152 . . . . . . 7 ((𝐹𝑥) ∈ 𝐵 → ((𝐹𝑥) = 𝑦𝑦𝐵))
1310, 12syl6 33 . . . . . 6 (∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵 → (𝑥𝐴 → ((𝐹𝑥) = 𝑦𝑦𝐵)))
148, 9, 13rexlimd 2447 . . . . 5 (∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵 → (∃𝑥𝐴 (𝐹𝑥) = 𝑦𝑦𝐵))
157, 14sylan9 395 . . . 4 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → (𝑦 ∈ ran 𝐹𝑦𝐵))
1615ssrdv 2978 . . 3 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → ran 𝐹𝐵)
17 df-f 4933 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
185, 16, 17sylanbrc 402 . 2 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → 𝐹:𝐴𝐵)
194, 18impbii 121 1 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  wral 2323  wrex 2324  wss 2944  ran crn 4373   Fn wfn 4924  wf 4925  cfv 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-fv 4937
This theorem is referenced by:  ffnfvf  5351  fnfvrnss  5352  fmpt2d  5354  ffnov  5632  cnref1o  8679  iseqf  9387  shftf  9658
  Copyright terms: Public domain W3C validator