Theorem dfafn5b 47787

Description: Representation of a function in terms of its values, analogous to dffn5 6940 (only if it is assumed that the function value for each x is a set). (Contributed by Alexander van der Vekens, 25-May-2017.)

Assertion

Ref	Expression
dfafn5b	⊢ (∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝑉 → (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥))))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem dfafn5b

Step	Hyp	Ref	Expression
1		dfafn5a 47786	. 2 ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)))
2		eqid 2769	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥))
3	2	fnmpt 6676	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)) Fn 𝐴)
4		fneq1 6627	. . 3 ⊢ (𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)) → (𝐹 Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)) Fn 𝐴))
5	3, 4	syl5ibrcom 250	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝑉 → (𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)) → 𝐹 Fn 𝐴))
6	1, 5	impbid2 229	1 ⊢ (∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝑉 → (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥))))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1567   ∈ wcel 2149  ∀wral 3085   ↦ cmpt 5196   Fn wfn 6532  '''cafv 47743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-res 5674  df-iota 6493  df-fun 6539  df-fn 6540  df-fv 6545  df-aiota 47711  df-dfat 47745  df-afv 47746

This theorem is referenced by: (None)