Theorem f1o00 5496

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 5469	. 2 ⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴))
2		fn0 5335	. . . . . 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 4801	. . . . . . . . . 10 ⊢ (𝐹 = ∅ → ◡𝐹 = ◡∅)
6		cnv0 5032	. . . . . . . . . 10 ⊢ ◡∅ = ∅
7	5, 6	eqtrdi 2226	. . . . . . . . 9 ⊢ (𝐹 = ∅ → ◡𝐹 = ∅)
8	2, 7	sylbi 121	. . . . . . . 8 ⊢ (𝐹 Fn ∅ → ◡𝐹 = ∅)
9	8	fneq1d 5306	. . . . . . 7 ⊢ (𝐹 Fn ∅ → (◡𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴))
10	9	biimpa 296	. . . . . 6 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → ∅ Fn 𝐴)
11		fndm 5315	. . . . . 6 ⊢ (∅ Fn 𝐴 → dom ∅ = 𝐴)
12	10, 11	syl 14	. . . . 5 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → dom ∅ = 𝐴)
13		dm0 4841	. . . . 5 ⊢ dom ∅ = ∅
14	12, 13	eqtr3di 2225	. . . 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 2177	. . . . . 6 ⊢ ∅ = ∅
19		fn0 5335	. . . . . 6 ⊢ (∅ Fn ∅ ↔ ∅ = ∅)
20	18, 19	mpbir 146	. . . . 5 ⊢ ∅ Fn ∅
21	7	fneq1d 5306	. . . . . 6 ⊢ (𝐹 = ∅ → (◡𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴))
22		fneq2 5305	. . . . . 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 1353  ∅c0 3422  ^◡ccnv 4625  dom cdm 4626   Fn wfn 5211  –1-1-onto→wf1o 5215

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223

This theorem is referenced by:  fo00  5497  f1o0  5498  en0  6794