Theorem fintm 5443

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 3890	. . . 4 ⊢ (ran 𝐹 ⊆ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 ran 𝐹 ⊆ 𝑥)
2	1	anbi2i 457	. . 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 3543	. . . 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 187	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ∩ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝑥))
7		df-f 5262	. 2 ⊢ (𝐹:𝐴⟶∩ 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ∩ 𝐵))
8		df-f 5262	. . 3 ⊢ (𝐹:𝐴⟶𝑥 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝑥))
9	8	ralbii 2503	. 2 ⊢ (∀𝑥 ∈ 𝐵 𝐹:𝐴⟶𝑥 ↔ ∀𝑥 ∈ 𝐵 (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝑥))
10	6, 7, 9	3bitr4i 212	1 ⊢ (𝐹:𝐴⟶∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐹:𝐴⟶𝑥)

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃wex 1506   ∈ wcel 2167  ∀wral 2475   ⊆ wss 3157  ∩ cint 3874  ran crn 4664   Fn wfn 5253  ⟶wf 5254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-in 3163  df-ss 3170  df-int 3875  df-f 5262

This theorem is referenced by: (None)