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Theorem ffnafv 41572
Description: A function maps to a class to which all values belong, analogous to ffnfv 6428. (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 6083 . . 3 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 fafvelrn 41571 . . . 4 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹'''𝑥) ∈ 𝐵)
32ralrimiva 2995 . . 3 (𝐹:𝐴𝐵 → ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵)
41, 3jca 553 . 2 (𝐹:𝐴𝐵 → (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵))
5 simpl 472 . . 3 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵) → 𝐹 Fn 𝐴)
6 afvelrnb0 41565 . . . . 5 (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 → ∃𝑥𝐴 (𝐹'''𝑥) = 𝑦))
7 nfra1 2970 . . . . . 6 𝑥𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵
8 nfv 1883 . . . . . 6 𝑥 𝑦𝐵
9 rsp 2958 . . . . . . 7 (∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵 → (𝑥𝐴 → (𝐹'''𝑥) ∈ 𝐵))
10 eleq1 2718 . . . . . . . 8 ((𝐹'''𝑥) = 𝑦 → ((𝐹'''𝑥) ∈ 𝐵𝑦𝐵))
1110biimpcd 239 . . . . . . 7 ((𝐹'''𝑥) ∈ 𝐵 → ((𝐹'''𝑥) = 𝑦𝑦𝐵))
129, 11syl6 35 . . . . . 6 (∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵 → (𝑥𝐴 → ((𝐹'''𝑥) = 𝑦𝑦𝐵)))
137, 8, 12rexlimd 3055 . . . . 5 (∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵 → (∃𝑥𝐴 (𝐹'''𝑥) = 𝑦𝑦𝐵))
146, 13sylan9 690 . . . 4 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵) → (𝑦 ∈ ran 𝐹𝑦𝐵))
1514ssrdv 3642 . . 3 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵) → ran 𝐹𝐵)
16 df-f 5930 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
175, 15, 16sylanbrc 699 . 2 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵) → 𝐹:𝐴𝐵)
184, 17impbii 199 1 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  wss 3607  ran crn 5144   Fn wfn 5921  wf 5922  '''cafv 41515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-dfat 41517  df-afv 41518
This theorem is referenced by:  ffnaov  41600
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