Theorem f1o00 5104

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 5077	. 2 ⊢ (𝐹:∅–1-1-onto→A ↔ (𝐹 Fn ∅ ∧ ◡𝐹 Fn A))
2		fn0 4961	. . . . . 6 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅)
3	2	biimpi 113	. . . . 5 ⊢ (𝐹 Fn ∅ → 𝐹 = ∅)
4	3	adantr 261	. . . 4 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn A) → 𝐹 = ∅)
5		dm0 4492	. . . . 5 ⊢ dom ∅ = ∅
6		cnveq 4452	. . . . . . . . . 10 ⊢ (𝐹 = ∅ → ◡𝐹 = ◡∅)
7		cnv0 4670	. . . . . . . . . 10 ⊢ ◡∅ = ∅
8	6, 7	syl6eq 2085	. . . . . . . . 9 ⊢ (𝐹 = ∅ → ◡𝐹 = ∅)
9	2, 8	sylbi 114	. . . . . . . 8 ⊢ (𝐹 Fn ∅ → ◡𝐹 = ∅)
10	9	fneq1d 4932	. . . . . . 7 ⊢ (𝐹 Fn ∅ → (◡𝐹 Fn A ↔ ∅ Fn A))
11	10	biimpa 280	. . . . . 6 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn A) → ∅ Fn A)
12		fndm 4941	. . . . . 6 ⊢ (∅ Fn A → dom ∅ = A)
13	11, 12	syl 14	. . . . 5 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn A) → dom ∅ = A)
14	5, 13	syl5reqr 2084	. . . 4 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn A) → A = ∅)
15	4, 14	jca 290	. . 3 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn A) → (𝐹 = ∅ ∧ A = ∅))
16	2	biimpri 124	. . . . 5 ⊢ (𝐹 = ∅ → 𝐹 Fn ∅)
17	16	adantr 261	. . . 4 ⊢ ((𝐹 = ∅ ∧ A = ∅) → 𝐹 Fn ∅)
18		eqid 2037	. . . . . 6 ⊢ ∅ = ∅
19		fn0 4961	. . . . . 6 ⊢ (∅ Fn ∅ ↔ ∅ = ∅)
20	18, 19	mpbir 134	. . . . 5 ⊢ ∅ Fn ∅
21	8	fneq1d 4932	. . . . . 6 ⊢ (𝐹 = ∅ → (◡𝐹 Fn A ↔ ∅ Fn A))
22		fneq2 4931	. . . . . 6 ⊢ (A = ∅ → (∅ Fn A ↔ ∅ Fn ∅))
23	21, 22	sylan9bb 435	. . . . 5 ⊢ ((𝐹 = ∅ ∧ A = ∅) → (◡𝐹 Fn A ↔ ∅ Fn ∅))
24	20, 23	mpbiri 157	. . . 4 ⊢ ((𝐹 = ∅ ∧ A = ∅) → ◡𝐹 Fn A)
25	17, 24	jca 290	. . 3 ⊢ ((𝐹 = ∅ ∧ A = ∅) → (𝐹 Fn ∅ ∧ ◡𝐹 Fn A))
26	15, 25	impbii 117	. 2 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn A) ↔ (𝐹 = ∅ ∧ A = ∅))
27	1, 26	bitri 173	1 ⊢ (𝐹:∅–1-1-onto→A ↔ (𝐹 = ∅ ∧ A = ∅))

Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1242  ∅c0 3218  ^◡ccnv 4287  dom cdm 4288   Fn wfn 4840  –1-1-onto→wf1o 4844

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852

This theorem is referenced by:  fo00  5105  f1o0  5106  en0  6211