Theorem funssxp 3652

Description: Two ways of specifying a partial function from A to B.

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 3556	. . . . . 6 ⊢ (Fun F ↔ F Fn dom F)
2	1	biimp 151	. . . . 5 ⊢ (Fun F → F Fn dom F)
3		rnss 3356	. . . . . 6 ⊢ (F ⊆ (A × B) → ran F ⊆ ran ( A × B))
4		rnxpss 3488	. . . . . . 7 ⊢ ran ( A × B) ⊆ B
5		sstr 2081	. . . . . . 7 ⊢ ((ran F ⊆ ran ( A × B) ⋀ ran ( A × B) ⊆ B) → ran F ⊆ B)
6	4, 5	mpan2 700	. . . . . 6 ⊢ (ran F ⊆ ran ( A × B) → ran F ⊆ B)
7	3, 6	syl 10	. . . . 5 ⊢ (F ⊆ (A × B) → ran F ⊆ B)
8	2, 7	anim12i 333	. . . 4 ⊢ ((Fun F ⋀ F ⊆ (A × B)) → (F Fn dom F ⋀ ran F ⊆ B))
9		df-f 3208	. . . 4 ⊢ (F:dom F–→B ↔ (F Fn dom F ⋀ ran F ⊆ B))
10	8, 9	sylibr 200	. . 3 ⊢ ((Fun F ⋀ F ⊆ (A × B)) → F:dom F–→B)
11		dmss 3324	. . . . 5 ⊢ (F ⊆ (A × B) → dom F ⊆ dom ( A × B))
12		dmxpss 3487	. . . . . 6 ⊢ dom ( A × B) ⊆ A
13		sstr 2081	. . . . . 6 ⊢ ((dom F ⊆ dom ( A × B) ⋀ dom ( A × B) ⊆ A) → dom F ⊆ A)
14	12, 13	mpan2 700	. . . . 5 ⊢ (dom F ⊆ dom ( A × B) → dom F ⊆ A)
15	11, 14	syl 10	. . . 4 ⊢ (F ⊆ (A × B) → dom F ⊆ A)
16	15	adantl 390	. . 3 ⊢ ((Fun F ⋀ F ⊆ (A × B)) → dom F ⊆ A)
17	10, 16	jca 288	. 2 ⊢ ((Fun F ⋀ F ⊆ (A × B)) → (F:dom F–→B ⋀ dom F ⊆ A))
18		ffun 3643	. . . 4 ⊢ (F:dom F–→B → Fun F)
19	18	adantr 391	. . 3 ⊢ ((F:dom F–→B ⋀ dom F ⊆ A) → Fun F)
20		fssxp 3651	. . . 4 ⊢ (F:dom F–→B → F ⊆ (dom F × B))
21		ssid 2089	. . . . 5 ⊢ B ⊆ B
22		ssxp 3270	. . . . 5 ⊢ ((dom F ⊆ A ⋀ B ⊆ B) → (dom F × B) ⊆ (A × B))
23	21, 22	mpan2 700	. . . 4 ⊢ (dom F ⊆ A → (dom F × B) ⊆ (A × B))
24	20, 23	sylan9ss 2084	. . 3 ⊢ ((F:dom F–→B ⋀ dom F ⊆ A) → F ⊆ (A × B))
25	19, 24	jca 288	. 2 ⊢ ((F:dom F–→B ⋀ dom F ⊆ A) → (Fun F ⋀ F ⊆ (A × B)))
26	17, 25	impbi 157	1 ⊢ ((Fun F ⋀ F ⊆ (A × B)) ↔ (F:dom F–→B ⋀ dom F ⊆ A))

Syntax hints:   ↔ wb 146   ⋀ wa 223   ⊆ wss 2056   × cxp 3182  dom cdm 3184  ran crn 3185  Fun wfun 3190   Fn wfn 3191  –→wf 3192

This theorem is referenced by:  elpm2 4351

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-9 969  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1129  ax-10o 1146  ax-16 1216  ax-11o 1224  ax-ext 1466  ax-sep 2716  ax-pow 2756  ax-pr 2793

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 985  df-sb 1178  df-eu 1388  df-mo 1389  df-clab 1471  df-cleq 1476  df-clel 1479  df-ne 1594  df-ral 1656  df-v 1819  df-dif 2058  df-un 2059  df-in 2060  df-ss 2062  df-nul 2290  df-pw 2412  df-sn 2422  df-pr 2423  df-op 2426  df-br 2633  df-opab 2680  df-xp 3198  df-rel 3199  df-cnv 3200  df-dm 3202  df-rn 3203  df-fun 3206  df-fn 3207  df-f 3208

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