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Theorem ffnafv 42913
Description: A function maps to a class to which all values belong, analogous to ffnfv 6750. (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 6387 . . 3 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 fafvelrn 42912 . . . 4 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹'''𝑥) ∈ 𝐵)
32ralrimiva 3149 . . 3 (𝐹:𝐴𝐵 → ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵)
41, 3jca 512 . 2 (𝐹:𝐴𝐵 → (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵))
5 simpl 483 . . 3 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵) → 𝐹 Fn 𝐴)
6 afvelrnb0 42906 . . . . 5 (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 → ∃𝑥𝐴 (𝐹'''𝑥) = 𝑦))
7 nfra1 3186 . . . . . 6 𝑥𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵
8 nfv 1892 . . . . . 6 𝑥 𝑦𝐵
9 rsp 3172 . . . . . . 7 (∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵 → (𝑥𝐴 → (𝐹'''𝑥) ∈ 𝐵))
10 eleq1 2870 . . . . . . . 8 ((𝐹'''𝑥) = 𝑦 → ((𝐹'''𝑥) ∈ 𝐵𝑦𝐵))
1110biimpcd 250 . . . . . . 7 ((𝐹'''𝑥) ∈ 𝐵 → ((𝐹'''𝑥) = 𝑦𝑦𝐵))
129, 11syl6 35 . . . . . 6 (∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵 → (𝑥𝐴 → ((𝐹'''𝑥) = 𝑦𝑦𝐵)))
137, 8, 12rexlimd 3278 . . . . 5 (∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵 → (∃𝑥𝐴 (𝐹'''𝑥) = 𝑦𝑦𝐵))
146, 13sylan9 508 . . . 4 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵) → (𝑦 ∈ ran 𝐹𝑦𝐵))
1514ssrdv 3899 . . 3 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵) → ran 𝐹𝐵)
16 df-f 6234 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
175, 15, 16sylanbrc 583 . 2 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵) → 𝐹:𝐴𝐵)
184, 17impbii 210 1 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1522  wcel 2081  wral 3105  wrex 3106  wss 3863  ran crn 5449   Fn wfn 6225  wf 6226  '''cafv 42859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-nul 4216  df-if 4386  df-sn 4477  df-pr 4479  df-op 4483  df-uni 4750  df-int 4787  df-br 4967  df-opab 5029  df-mpt 5046  df-id 5353  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-fv 6238  df-aiota 42828  df-dfat 42861  df-afv 42862
This theorem is referenced by:  ffnaov  42941
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