Theorem f1o00 2823

Description: One-to-one onto mapping of the empty set.

Assertion

Ref	Expression
f1o00	⊢ (F:∅–1-1-onto→A ↔ (F = ∅ ∧ A = ∅))

Proof of Theorem f1o00

Step	Hyp	Ref	Expression
1		f1o4 2807	. 2 ⊢ (F:∅–1-1-onto→A ↔ (F Fn ∅ ∧ ◡F Fn A))
2		fn0 2739	. . . . . 6 ⊢ (F Fn ∅ ↔ F = ∅)
3	2	biimp 133	. . . . 5 ⊢ (F Fn ∅ → F = ∅)
4	3	adantr 306	. . . 4 ⊢ ((F Fn ∅ ∧ ◡F Fn A) → F = ∅)
5		cnveq 2513	. . . . . . . . . 10 ⊢ (F = ∅ → ◡F = ◡∅)
6		cnv0 2633	. . . . . . . . . 10 ⊢ ◡∅ = ∅
7	5, 6	syl6eq 1140	. . . . . . . . 9 ⊢ (F = ∅ → ◡F = ∅)
8	2, 7	sylbi 174	. . . . . . . 8 ⊢ (F Fn ∅ → ◡F = ∅)
9		fneq1 2718	. . . . . . . 8 ⊢ (◡F = ∅ → (◡F Fn A ↔ ∅ Fn A))
10	8, 9	syl 12	. . . . . . 7 ⊢ (F Fn ∅ → (◡F Fn A ↔ ∅ Fn A))
11	10	biimpa 324	. . . . . 6 ⊢ ((F Fn ∅ ∧ ◡F Fn A) → ∅ Fn A)
12		fndm 2723	. . . . . 6 ⊢ (∅ Fn A → dom ∅ = A)
13	11, 12	syl 12	. . . . 5 ⊢ ((F Fn ∅ ∧ ◡F Fn A) → dom ∅ = A)
14		dm0 2542	. . . . 5 ⊢ dom ∅ = ∅
15	13, 14	syl5reqr 1139	. . . 4 ⊢ ((F Fn ∅ ∧ ◡F Fn A) → A = ∅)
16	4, 15	jca 236	. . 3 ⊢ ((F Fn ∅ ∧ ◡F Fn A) → (F = ∅ ∧ A = ∅))
17	2	biimpr 134	. . . . 5 ⊢ (F = ∅ → F Fn ∅)
18	17	adantr 306	. . . 4 ⊢ ((F = ∅ ∧ A = ∅) → F Fn ∅)
19		cleqid 1102	. . . . . 6 ⊢ ∅ = ∅
20		fn0 2739	. . . . . 6 ⊢ (∅ Fn ∅ ↔ ∅ = ∅)
21	19, 20	mpbir 165	. . . . 5 ⊢ ∅ Fn ∅
22	7, 9	syl 12	. . . . . 6 ⊢ (F = ∅ → (◡F Fn A ↔ ∅ Fn A))
23		fneq2 2719	. . . . . 6 ⊢ (A = ∅ → (∅ Fn A ↔ ∅ Fn ∅))
24	22, 23	sylan9bb 418	. . . . 5 ⊢ ((F = ∅ ∧ A = ∅) → (◡F Fn A ↔ ∅ Fn ∅))
25	21, 24	mpbiri 169	. . . 4 ⊢ ((F = ∅ ∧ A = ∅) → ◡F Fn A)
26	18, 25	jca 236	. . 3 ⊢ ((F = ∅ ∧ A = ∅) → (F Fn ∅ ∧ ◡F Fn A))
27	16, 26	impbi 139	. 2 ⊢ ((F Fn ∅ ∧ ◡F Fn A) ↔ (F = ∅ ∧ A = ∅))
28	1, 27	bitr 151	1 ⊢ (F:∅–1-1-onto→A ↔ (F = ∅ ∧ A = ∅))

Syntax hints:   ↔ wb 127   ∧ wa 196   = wceq 1091  ∅c0 1707  ^◡ccnv 2409  dom cdm 2410   Fn wfn 2417  –1-1-onto→wf1o 2421

This theorem is referenced by:  f1o0 2824  en0 3328

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-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  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-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437

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