Theorem f1o00 5536

Description: One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.)

Assertion

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

Proof of Theorem f1o00

Step	Hyp	Ref	Expression
1		dff1o4 5509	. 2 ⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴))
2		fn0 5374	. . . . . 6 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅)
3	2	biimpi 120	. . . . 5 ⊢ (𝐹 Fn ∅ → 𝐹 = ∅)
4	3	adantr 276	. . . 4 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → 𝐹 = ∅)
5		cnveq 4837	. . . . . . . . . 10 ⊢ (𝐹 = ∅ → ◡𝐹 = ◡∅)
6		cnv0 5070	. . . . . . . . . 10 ⊢ ◡∅ = ∅
7	5, 6	eqtrdi 2242	. . . . . . . . 9 ⊢ (𝐹 = ∅ → ◡𝐹 = ∅)
8	2, 7	sylbi 121	. . . . . . . 8 ⊢ (𝐹 Fn ∅ → ◡𝐹 = ∅)
9	8	fneq1d 5345	. . . . . . 7 ⊢ (𝐹 Fn ∅ → (◡𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴))
10	9	biimpa 296	. . . . . 6 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → ∅ Fn 𝐴)
11		fndm 5354	. . . . . 6 ⊢ (∅ Fn 𝐴 → dom ∅ = 𝐴)
12	10, 11	syl 14	. . . . 5 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → dom ∅ = 𝐴)
13		dm0 4877	. . . . 5 ⊢ dom ∅ = ∅
14	12, 13	eqtr3di 2241	. . . 4 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → 𝐴 = ∅)
15	4, 14	jca 306	. . 3 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → (𝐹 = ∅ ∧ 𝐴 = ∅))
16	2	biimpri 133	. . . . 5 ⊢ (𝐹 = ∅ → 𝐹 Fn ∅)
17	16	adantr 276	. . . 4 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹 Fn ∅)
18		eqid 2193	. . . . . 6 ⊢ ∅ = ∅
19		fn0 5374	. . . . . 6 ⊢ (∅ Fn ∅ ↔ ∅ = ∅)
20	18, 19	mpbir 146	. . . . 5 ⊢ ∅ Fn ∅
21	7	fneq1d 5345	. . . . . 6 ⊢ (𝐹 = ∅ → (◡𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴))
22		fneq2 5344	. . . . . 6 ⊢ (𝐴 = ∅ → (∅ Fn 𝐴 ↔ ∅ Fn ∅))
23	21, 22	sylan9bb 462	. . . . 5 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → (◡𝐹 Fn 𝐴 ↔ ∅ Fn ∅))
24	20, 23	mpbiri 168	. . . 4 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → ◡𝐹 Fn 𝐴)
25	17, 24	jca 306	. . 3 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴))
26	15, 25	impbii 126	. 2 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
27	1, 26	bitri 184	1 ⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1364  ∅c0 3447  ^◡ccnv 4659  dom cdm 4660   Fn wfn 5250  –1-1-onto→wf1o 5254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262

This theorem is referenced by:  fo00  5537  f1o0  5538  en0  6851