Theorem f1o00 6171

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 6145	. 2 ⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴))
2		fn0 6011	. . . . . 6 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅)
3	2	biimpi 206	. . . . 5 ⊢ (𝐹 Fn ∅ → 𝐹 = ∅)
4	3	adantr 481	. . . 4 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → 𝐹 = ∅)
5		dm0 5339	. . . . 5 ⊢ dom ∅ = ∅
6		cnveq 5296	. . . . . . . . . 10 ⊢ (𝐹 = ∅ → ◡𝐹 = ◡∅)
7		cnv0 5535	. . . . . . . . . 10 ⊢ ◡∅ = ∅
8	6, 7	syl6eq 2672	. . . . . . . . 9 ⊢ (𝐹 = ∅ → ◡𝐹 = ∅)
9	2, 8	sylbi 207	. . . . . . . 8 ⊢ (𝐹 Fn ∅ → ◡𝐹 = ∅)
10	9	fneq1d 5981	. . . . . . 7 ⊢ (𝐹 Fn ∅ → (◡𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴))
11	10	biimpa 501	. . . . . 6 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → ∅ Fn 𝐴)
12		fndm 5990	. . . . . 6 ⊢ (∅ Fn 𝐴 → dom ∅ = 𝐴)
13	11, 12	syl 17	. . . . 5 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → dom ∅ = 𝐴)
14	5, 13	syl5reqr 2671	. . . 4 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → 𝐴 = ∅)
15	4, 14	jca 554	. . 3 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → (𝐹 = ∅ ∧ 𝐴 = ∅))
16	2	biimpri 218	. . . . 5 ⊢ (𝐹 = ∅ → 𝐹 Fn ∅)
17	16	adantr 481	. . . 4 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹 Fn ∅)
18		eqid 2622	. . . . . 6 ⊢ ∅ = ∅
19		fn0 6011	. . . . . 6 ⊢ (∅ Fn ∅ ↔ ∅ = ∅)
20	18, 19	mpbir 221	. . . . 5 ⊢ ∅ Fn ∅
21	8	fneq1d 5981	. . . . . 6 ⊢ (𝐹 = ∅ → (◡𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴))
22		fneq2 5980	. . . . . 6 ⊢ (𝐴 = ∅ → (∅ Fn 𝐴 ↔ ∅ Fn ∅))
23	21, 22	sylan9bb 736	. . . . 5 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → (◡𝐹 Fn 𝐴 ↔ ∅ Fn ∅))
24	20, 23	mpbiri 248	. . . 4 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → ◡𝐹 Fn 𝐴)
25	17, 24	jca 554	. . 3 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴))
26	15, 25	impbii 199	. 2 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
27	1, 26	bitri 264	1 ⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1483  ∅c0 3915  ^◡ccnv 5113  dom cdm 5114   Fn wfn 5883  –1-1-onto→wf1o 5887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895

This theorem is referenced by:  fo00  6172  f1o0  6173  en0  8019  symgbas0  17814  derang0  31151  poimirlem28  33437