Theorem fintm 5352

Description: Function into an intersection. (Contributed by Jim Kingdon, 28-Dec-2018.)

Hypothesis

Ref	Expression
fintm.1	⊢ ∃𝑥 𝑥 ∈ 𝐵

Assertion

Ref	Expression
fintm	⊢ (𝐹:𝐴⟶∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐹:𝐴⟶𝑥)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Proof of Theorem fintm

Step	Hyp	Ref	Expression
1		ssint 3823	. . . 4 ⊢ (ran 𝐹 ⊆ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 ran 𝐹 ⊆ 𝑥)
2	1	anbi2i 453	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ∩ 𝐵) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐵 ran 𝐹 ⊆ 𝑥))
3		fintm.1	. . . 4 ⊢ ∃𝑥 𝑥 ∈ 𝐵
4		r19.28mv 3486	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝑥) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐵 ran 𝐹 ⊆ 𝑥)))
5	3, 4	ax-mp 5	. . 3 ⊢ (∀𝑥 ∈ 𝐵 (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝑥) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐵 ran 𝐹 ⊆ 𝑥))
6	2, 5	bitr4i 186	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ∩ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝑥))
7		df-f 5171	. 2 ⊢ (𝐹:𝐴⟶∩ 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ∩ 𝐵))
8		df-f 5171	. . 3 ⊢ (𝐹:𝐴⟶𝑥 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝑥))
9	8	ralbii 2463	. 2 ⊢ (∀𝑥 ∈ 𝐵 𝐹:𝐴⟶𝑥 ↔ ∀𝑥 ∈ 𝐵 (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝑥))
10	6, 7, 9	3bitr4i 211	1 ⊢ (𝐹:𝐴⟶∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐹:𝐴⟶𝑥)

Syntax hints:   ∧ wa 103   ↔ wb 104  ∃wex 1472   ∈ wcel 2128  ∀wral 2435   ⊆ wss 3102  ∩ cint 3807  ran crn 4584   Fn wfn 5162  ⟶wf 5163

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-in 3108  df-ss 3115  df-int 3808  df-f 5171

This theorem is referenced by: (None)