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Theorem ffnafv 43770
 Description: A function maps to a class to which all values belong, analogous to ffnfv 6860. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
ffnafv (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Proof of Theorem ffnafv
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ffn 6488 . . 3 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 fafvelrn 43769 . . . 4 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹'''𝑥) ∈ 𝐵)
32ralrimiva 3149 . . 3 (𝐹:𝐴𝐵 → ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵)
41, 3jca 515 . 2 (𝐹:𝐴𝐵 → (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵))
5 simpl 486 . . 3 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵) → 𝐹 Fn 𝐴)
6 afvelrnb0 43763 . . . . 5 (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 → ∃𝑥𝐴 (𝐹'''𝑥) = 𝑦))
7 nfra1 3183 . . . . . 6 𝑥𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵
8 nfv 1915 . . . . . 6 𝑥 𝑦𝐵
9 rsp 3170 . . . . . . 7 (∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵 → (𝑥𝐴 → (𝐹'''𝑥) ∈ 𝐵))
10 eleq1 2877 . . . . . . . 8 ((𝐹'''𝑥) = 𝑦 → ((𝐹'''𝑥) ∈ 𝐵𝑦𝐵))
1110biimpcd 252 . . . . . . 7 ((𝐹'''𝑥) ∈ 𝐵 → ((𝐹'''𝑥) = 𝑦𝑦𝐵))
129, 11syl6 35 . . . . . 6 (∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵 → (𝑥𝐴 → ((𝐹'''𝑥) = 𝑦𝑦𝐵)))
137, 8, 12rexlimd 3276 . . . . 5 (∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵 → (∃𝑥𝐴 (𝐹'''𝑥) = 𝑦𝑦𝐵))
146, 13sylan9 511 . . . 4 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵) → (𝑦 ∈ ran 𝐹𝑦𝐵))
1514ssrdv 3921 . . 3 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵) → ran 𝐹𝐵)
16 df-f 6329 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
175, 15, 16sylanbrc 586 . 2 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵) → 𝐹:𝐴𝐵)
184, 17impbii 212 1 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ⊆ wss 3881  ran crn 5521   Fn wfn 6320  ⟶wf 6321  '''cafv 43716 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-int 4840  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-fv 6333  df-aiota 43685  df-dfat 43718  df-afv 43719 This theorem is referenced by:  ffnaov  43798
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