Theorem dff2 7097

Description: Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.)

Assertion

Ref	Expression
dff2	⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ 𝐹 ⊆ (𝐴 × 𝐵)))

Proof of Theorem dff2

Step	Hyp	Ref	Expression
1		ffn 6707	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
2		fssxp 6735	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵))
3	1, 2	jca 521	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 Fn 𝐴 ∧ 𝐹 ⊆ (𝐴 × 𝐵)))
4		rnss 5921	. . . . 5 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ ran (𝐴 × 𝐵))
5		rnxpss 6164	. . . . 5 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
6	4, 5	sstrdi 3943	. . . 4 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ 𝐵)
7	6	anim2i 629	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
8		df-f 6541	. . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
9	7, 8	sylibr 237	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → 𝐹:𝐴⟶𝐵)
10	3, 9	impbii 212	1 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ 𝐹 ⊆ (𝐴 × 𝐵)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ⊆ wss 3899   × cxp 5649  ran crn 5652   Fn wfn 6532  ⟶wf 6533

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658  df-cnv 5659  df-dm 5661  df-rn 5662  df-fun 6539  df-fn 6540  df-f 6541

This theorem is used by:  fpr2g  7215  mapval2  8893  cardf2  10017  mpoaddf  11287  mpomulf  11288  imasaddflem  17695  imasvscaf  17704  gsumpart  33617