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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfafn5b | Structured version Visualization version GIF version |
Description: Representation of a function in terms of its values, analogous to dffn5 6961 (only if it is assumed that the function value for each x is a set). (Contributed by Alexander van der Vekens, 25-May-2017.) |
Ref | Expression |
---|---|
dfafn5b | ⊢ (∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝑉 → (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfafn5a 46773 | . 2 ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥))) | |
2 | eqid 2726 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)) | |
3 | 2 | fnmpt 6701 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)) Fn 𝐴) |
4 | fneq1 6651 | . . 3 ⊢ (𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)) → (𝐹 Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)) Fn 𝐴)) | |
5 | 3, 4 | syl5ibrcom 246 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝑉 → (𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)) → 𝐹 Fn 𝐴)) |
6 | 1, 5 | impbid2 225 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝑉 → (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ↦ cmpt 5236 Fn wfn 6549 '''cafv 46730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-res 5694 df-iota 6506 df-fun 6556 df-fn 6557 df-fv 6562 df-aiota 46698 df-dfat 46732 df-afv 46733 |
This theorem is referenced by: (None) |
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