Theorem fssxp 2761

Description: A mapping is a class of ordered pairs.

Assertion

Ref	Expression
fssxp	⊢ (F:A–→B → F ⊆ (A × B))

Proof of Theorem fssxp

Step	Hyp	Ref	Expression
1		frn 2757	. . . 4 ⊢ (F:A–→B → ran F ⊆ B)
2		ssid 1519	. . . . 5 ⊢ A ⊆ A
3		ssxp 2487	. . . . 5 ⊢ ((A ⊆ A ∧ ran F ⊆ B) → (A × ran F) ⊆ (A × B))
4	2, 3	mpan 518	. . . 4 ⊢ (ran F ⊆ B → (A × ran F) ⊆ (A × B))
5	1, 4	syl 12	. . 3 ⊢ (F:A–→B → (A × ran F) ⊆ (A × B))
6		fdm 2756	. . . 4 ⊢ (F:A–→B → dom F = A)
7		xpeq1 2440	. . . 4 ⊢ (dom F = A → (dom F × ran F) = (A × ran F))
8		sseq1 1521	. . . 4 ⊢ ((dom F × ran F) = (A × ran F) → ((dom F × ran F) ⊆ (A × B) ↔ (A × ran F) ⊆ (A × B)))
9	6, 7, 8	3syl 21	. . 3 ⊢ (F:A–→B → ((dom F × ran F) ⊆ (A × B) ↔ (A × ran F) ⊆ (A × B)))
10	5, 9	mpbird 171	. 2 ⊢ (F:A–→B → (dom F × ran F) ⊆ (A × B))
11		frel 2755	. . 3 ⊢ (F:A–→B → Rel F)
12		relssdr 2668	. . 3 ⊢ (Rel F → F ⊆ (dom F × ran F))
13		sstr2 1510	. . 3 ⊢ (F ⊆ (dom F × ran F) → ((dom F × ran F) ⊆ (A × B) → F ⊆ (A × B)))
14	11, 12, 13	3syl 21	. 2 ⊢ (F:A–→B → ((dom F × ran F) ⊆ (A × B) → F ⊆ (A × B)))
15	10, 14	mpd 46	1 ⊢ (F:A–→B → F ⊆ (A × B))

Syntax hints:   → wi 2   ↔ wb 127   = wceq 1091   ⊆ wss 1487   × cxp 2408  dom cdm 2410  ran crn 2411  Rel wrel 2415  –→wf 2418

This theorem is referenced by:  opelf 2762  mapex 3261  infmap2 4953

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426  df-dm 2428  df-rn 2429  df-fun 2432  df-fn 2433  df-f 2434

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