Theorem funssxp 5974

Description: Two ways of specifying a partial function from 𝐴 to 𝐵. (Contributed by NM, 13-Nov-2007.)

Assertion

Ref	Expression
funssxp	⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) ↔ (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴))

Proof of Theorem funssxp

Step	Hyp	Ref	Expression
1		funfn 5833	. . . . . 6 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2	1	biimpi 205	. . . . 5 ⊢ (Fun 𝐹 → 𝐹 Fn dom 𝐹)
3		rnss 5275	. . . . . 6 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ ran (𝐴 × 𝐵))
4		rnxpss 5485	. . . . . 6 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
5	3, 4	syl6ss 3580	. . . . 5 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ 𝐵)
6	2, 5	anim12i 588	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ 𝐵))
7		df-f 5808	. . . 4 ⊢ (𝐹:dom 𝐹⟶𝐵 ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ 𝐵))
8	6, 7	sylibr 223	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → 𝐹:dom 𝐹⟶𝐵)
9		dmss 5245	. . . . 5 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹 ⊆ dom (𝐴 × 𝐵))
10		dmxpss 5484	. . . . 5 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴
11	9, 10	syl6ss 3580	. . . 4 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹 ⊆ 𝐴)
12	11	adantl 481	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → dom 𝐹 ⊆ 𝐴)
13	8, 12	jca 553	. 2 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴))
14		ffun 5961	. . . 4 ⊢ (𝐹:dom 𝐹⟶𝐵 → Fun 𝐹)
15	14	adantr 480	. . 3 ⊢ ((𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴) → Fun 𝐹)
16		fssxp 5973	. . . 4 ⊢ (𝐹:dom 𝐹⟶𝐵 → 𝐹 ⊆ (dom 𝐹 × 𝐵))
17		xpss1 5151	. . . 4 ⊢ (dom 𝐹 ⊆ 𝐴 → (dom 𝐹 × 𝐵) ⊆ (𝐴 × 𝐵))
18	16, 17	sylan9ss 3581	. . 3 ⊢ ((𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴) → 𝐹 ⊆ (𝐴 × 𝐵))
19	15, 18	jca 553	. 2 ⊢ ((𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴) → (Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)))
20	13, 19	impbii 198	1 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) ↔ (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴))

Syntax hints:   ↔ wb 195   ∧ wa 383   ⊆ wss 3540   × cxp 5036  dom cdm 5038  ran crn 5039  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-fun 5806  df-fn 5807  df-f 5808

This theorem is referenced by:  elpm2g  7760  volf  23104