Theorem funssxp 5367

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 5228	. . . . . 6 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2	1	biimpi 119	. . . . 5 ⊢ (Fun 𝐹 → 𝐹 Fn dom 𝐹)
3		rnss 4841	. . . . . 6 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ ran (𝐴 × 𝐵))
4		rnxpss 5042	. . . . . 6 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
5	3, 4	sstrdi 3159	. . . . 5 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ 𝐵)
6	2, 5	anim12i 336	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ 𝐵))
7		df-f 5202	. . . 4 ⊢ (𝐹:dom 𝐹⟶𝐵 ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ 𝐵))
8	6, 7	sylibr 133	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → 𝐹:dom 𝐹⟶𝐵)
9		dmss 4810	. . . . 5 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹 ⊆ dom (𝐴 × 𝐵))
10		dmxpss 5041	. . . . 5 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴
11	9, 10	sstrdi 3159	. . . 4 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹 ⊆ 𝐴)
12	11	adantl 275	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → dom 𝐹 ⊆ 𝐴)
13	8, 12	jca 304	. 2 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴))
14		ffun 5350	. . . 4 ⊢ (𝐹:dom 𝐹⟶𝐵 → Fun 𝐹)
15	14	adantr 274	. . 3 ⊢ ((𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴) → Fun 𝐹)
16		fssxp 5365	. . . 4 ⊢ (𝐹:dom 𝐹⟶𝐵 → 𝐹 ⊆ (dom 𝐹 × 𝐵))
17		xpss1 4721	. . . 4 ⊢ (dom 𝐹 ⊆ 𝐴 → (dom 𝐹 × 𝐵) ⊆ (𝐴 × 𝐵))
18	16, 17	sylan9ss 3160	. . 3 ⊢ ((𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴) → 𝐹 ⊆ (𝐴 × 𝐵))
19	15, 18	jca 304	. 2 ⊢ ((𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴) → (Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)))
20	13, 19	impbii 125	1 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) ↔ (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴))

Syntax hints:   ∧ wa 103   ↔ wb 104   ⊆ wss 3121   × cxp 4609  dom cdm 4611  ran crn 4612  Fun wfun 5192   Fn wfn 5193  ⟶wf 5194

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-dm 4621  df-rn 4622  df-fun 5200  df-fn 5201  df-f 5202

This theorem is referenced by:  elpm2g  6643  casef  7065