Theorem dfafn5b 48370

Description: Representation of a function in terms of its values, analogous to dffn5 6951 (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 48369	. 2 ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)))
2		eqid 2763	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥))
3	2	fnmpt 6687	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)) Fn 𝐴)
4		fneq1 6638	. . 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))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1569   ∈ wcel 2146  ∀wral 3079   ↦ cmpt 5188   Fn wfn 6541  '''cafv 48327

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-int 4910  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-res 5666  df-iota 6500  df-fun 6548  df-fn 6549  df-fv 6554  df-aiota 48294  df-dfat 48329  df-afv 48330

This theorem is used by: (None)