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Theorem ffnafv 45477
Description: A function maps to a class to which all values belong, analogous to ffnfv 7071. (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 6673 . . 3 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 fafvelcdm 45476 . . . 4 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹'''𝑥) ∈ 𝐵)
32ralrimiva 3144 . . 3 (𝐹:𝐴𝐵 → ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵)
41, 3jca 513 . 2 (𝐹:𝐴𝐵 → (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵))
5 simpl 484 . . 3 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵) → 𝐹 Fn 𝐴)
6 afvelrnb0 45470 . . . . 5 (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 → ∃𝑥𝐴 (𝐹'''𝑥) = 𝑦))
7 nfra1 3270 . . . . . 6 𝑥𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵
8 nfv 1918 . . . . . 6 𝑥 𝑦𝐵
9 rsp 3233 . . . . . . 7 (∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵 → (𝑥𝐴 → (𝐹'''𝑥) ∈ 𝐵))
10 eleq1 2826 . . . . . . . 8 ((𝐹'''𝑥) = 𝑦 → ((𝐹'''𝑥) ∈ 𝐵𝑦𝐵))
1110biimpcd 249 . . . . . . 7 ((𝐹'''𝑥) ∈ 𝐵 → ((𝐹'''𝑥) = 𝑦𝑦𝐵))
129, 11syl6 35 . . . . . 6 (∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵 → (𝑥𝐴 → ((𝐹'''𝑥) = 𝑦𝑦𝐵)))
137, 8, 12rexlimd 3252 . . . . 5 (∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵 → (∃𝑥𝐴 (𝐹'''𝑥) = 𝑦𝑦𝐵))
146, 13sylan9 509 . . . 4 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵) → (𝑦 ∈ ran 𝐹𝑦𝐵))
1514ssrdv 3955 . . 3 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵) → ran 𝐹𝐵)
16 df-f 6505 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
175, 15, 16sylanbrc 584 . 2 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵) → 𝐹:𝐴𝐵)
184, 17impbii 208 1 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3065  wrex 3074  wss 3915  ran crn 5639   Fn wfn 6496  wf 6497  '''cafv 45423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509  df-aiota 45391  df-dfat 45425  df-afv 45426
This theorem is referenced by:  ffnaov  45505
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