Theorem f1o00 6806

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 6779	. 2 ⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴))
2		fn0 6620	. . . . 5 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅)
3	2	birani 505	. . . 4 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → 𝐹 = ∅)
4		cnveq 5818	. . . . . . . . . 10 ⊢ (𝐹 = ∅ → ◡𝐹 = ◡∅)
5		cnv0 5828	. . . . . . . . . 10 ⊢ ◡∅ = ∅
6	4, 5	eqtrdi 2792	. . . . . . . . 9 ⊢ (𝐹 = ∅ → ◡𝐹 = ∅)
7	2, 6	sylbi 219	. . . . . . . 8 ⊢ (𝐹 Fn ∅ → ◡𝐹 = ∅)
8	7	fneq1d 6582	. . . . . . 7 ⊢ (𝐹 Fn ∅ → (◡𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴))
9	8	biimpa 478	. . . . . 6 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → ∅ Fn 𝐴)
10	9	fndmd 6594	. . . . 5 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → dom ∅ = 𝐴)
11		dm0 5869	. . . . 5 ⊢ dom ∅ = ∅
12	10, 11	eqtr3di 2791	. . . 4 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → 𝐴 = ∅)
13	3, 12	jca 517	. . 3 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → (𝐹 = ∅ ∧ 𝐴 = ∅))
14	2	biranri 507	. . . 4 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹 Fn ∅)
15		eqid 2741	. . . . . 6 ⊢ ∅ = ∅
16		fn0 6620	. . . . . 6 ⊢ (∅ Fn ∅ ↔ ∅ = ∅)
17	15, 16	mpbir 233	. . . . 5 ⊢ ∅ Fn ∅
18	6	fneq1d 6582	. . . . . 6 ⊢ (𝐹 = ∅ → (◡𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴))
19		fneq2 6581	. . . . . 6 ⊢ (𝐴 = ∅ → (∅ Fn 𝐴 ↔ ∅ Fn ∅))
20	18, 19	sylan9bb 515	. . . . 5 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → (◡𝐹 Fn 𝐴 ↔ ∅ Fn ∅))
21	17, 20	mpbiri 260	. . . 4 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → ◡𝐹 Fn 𝐴)
22	14, 21	jca 517	. . 3 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴))
23	13, 22	impbii 211	. 2 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
24	1, 23	bitri 277	1 ⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))

Syntax hints:   ↔ wb 208   ∧ wa 397   = wceq 1548  ∅c0 4264  ^◡ccnv 5620  dom cdm 5621   Fn wfn 6484  –1-1-onto→wf1o 6488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-mo 2545  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496

This theorem is referenced by:  fo00  6807  f1o0  6808  en0  8959  en0ALT  8960  en0r  8961  infn0  9206  symgbas0  19359  derang0  35412  poimirlem28  38030