Theorem funssxp 6726

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 6558	. . . . . 6 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2	1	biimpi 219	. . . . 5 ⊢ (Fun 𝐹 → 𝐹 Fn dom 𝐹)
3		rnss 5917	. . . . . 6 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ ran (𝐴 × 𝐵))
4		rnxpss 6159	. . . . . 6 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
5	3, 4	sstrdi 3942	. . . . 5 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ 𝐵)
6	2, 5	anim12i 625	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ 𝐵))
7		df-f 6531	. . . 4 ⊢ (𝐹:dom 𝐹⟶𝐵 ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ 𝐵))
8	6, 7	sylibr 237	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → 𝐹:dom 𝐹⟶𝐵)
9		dmss 5880	. . . . 5 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹 ⊆ dom (𝐴 × 𝐵))
10		dmxpss 6158	. . . . 5 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴
11	9, 10	sstrdi 3942	. . . 4 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹 ⊆ 𝐴)
12	11	adantl 487	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → dom 𝐹 ⊆ 𝐴)
13	8, 12	jca 521	. 2 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴))
14		ffun 6700	. . . 4 ⊢ (𝐹:dom 𝐹⟶𝐵 → Fun 𝐹)
15	14	adantr 486	. . 3 ⊢ ((𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴) → Fun 𝐹)
16		fssxp 6725	. . . 4 ⊢ (𝐹:dom 𝐹⟶𝐵 → 𝐹 ⊆ (dom 𝐹 × 𝐵))
17		xpss1 5666	. . . 4 ⊢ (dom 𝐹 ⊆ 𝐴 → (dom 𝐹 × 𝐵) ⊆ (𝐴 × 𝐵))
18	16, 17	sylan9ss 3943	. . 3 ⊢ ((𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴) → 𝐹 ⊆ (𝐴 × 𝐵))
19	15, 18	jca 521	. 2 ⊢ ((𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴) → (Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)))
20	13, 19	impbii 212	1 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) ↔ (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ⊆ wss 3898   × cxp 5645  dom cdm 5647  ran crn 5648  Fun wfun 6521   Fn wfn 6522  ⟶wf 6523

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 2732  ax-sep 5248  ax-pr 5390

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 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5653  df-rel 5654  df-cnv 5655  df-dm 5657  df-rn 5658  df-fun 6529  df-fn 6530  df-f 6531

This theorem is used by:  elpm2g  8842  volf  25812  dfno2  44372