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Theorem ffnafv 46432
Description: A function maps to a class to which all values belong, analogous to ffnfv 7113. (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 6710 . . 3 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 fafvelcdm 46431 . . . 4 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹'''𝑥) ∈ 𝐵)
32ralrimiva 3140 . . 3 (𝐹:𝐴𝐵 → ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵)
41, 3jca 511 . 2 (𝐹:𝐴𝐵 → (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵))
5 simpl 482 . . 3 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵) → 𝐹 Fn 𝐴)
6 afvelrnb0 46425 . . . . 5 (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 → ∃𝑥𝐴 (𝐹'''𝑥) = 𝑦))
7 nfra1 3275 . . . . . 6 𝑥𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵
8 nfv 1909 . . . . . 6 𝑥 𝑦𝐵
9 rsp 3238 . . . . . . 7 (∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵 → (𝑥𝐴 → (𝐹'''𝑥) ∈ 𝐵))
10 eleq1 2815 . . . . . . . 8 ((𝐹'''𝑥) = 𝑦 → ((𝐹'''𝑥) ∈ 𝐵𝑦𝐵))
1110biimpcd 248 . . . . . . 7 ((𝐹'''𝑥) ∈ 𝐵 → ((𝐹'''𝑥) = 𝑦𝑦𝐵))
129, 11syl6 35 . . . . . 6 (∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵 → (𝑥𝐴 → ((𝐹'''𝑥) = 𝑦𝑦𝐵)))
137, 8, 12rexlimd 3257 . . . . 5 (∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵 → (∃𝑥𝐴 (𝐹'''𝑥) = 𝑦𝑦𝐵))
146, 13sylan9 507 . . . 4 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵) → (𝑦 ∈ ran 𝐹𝑦𝐵))
1514ssrdv 3983 . . 3 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝐵) → ran 𝐹𝐵)
16 df-f 6540 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
175, 15, 16sylanbrc 582 . 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 395   = wceq 1533  wcel 2098  wral 3055  wrex 3064  wss 3943  ran crn 5670   Fn wfn 6531  wf 6532  '''cafv 46378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-aiota 46346  df-dfat 46380  df-afv 46381
This theorem is referenced by:  ffnaov  46460
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