Theorem dfafn5b 43367

Description: Representation of a function in terms of its values, analogous to dffn5 6726 (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 43366	. 2 ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)))
2		eqid 2823	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥))
3	2	fnmpt 6490	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)) Fn 𝐴)
4		fneq1 6446	. . 3 ⊢ (𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)) → (𝐹 Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)) Fn 𝐴))
5	3, 4	syl5ibrcom 249	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝑉 → (𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥)) → 𝐹 Fn 𝐴))
6	1, 5	impbid2 228	1 ⊢ (∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝑉 → (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹'''𝑥))))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1537   ∈ wcel 2114  ∀wral 3140   ↦ cmpt 5148   Fn wfn 6352  '''cafv 43323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-int 4879  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-res 5569  df-iota 6316  df-fun 6359  df-fn 6360  df-fv 6365  df-aiota 43292  df-dfat 43325  df-afv 43326

This theorem is referenced by: (None)