Theorem f1o00 6817

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 6790	. 2 ⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴))
2		fn0 6631	. . . . . 6 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅)
3	2	biimpi 216	. . . . 5 ⊢ (𝐹 Fn ∅ → 𝐹 = ∅)
4	3	adantr 480	. . . 4 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → 𝐹 = ∅)
5		cnveq 5830	. . . . . . . . . 10 ⊢ (𝐹 = ∅ → ◡𝐹 = ◡∅)
6		cnv0 6105	. . . . . . . . . 10 ⊢ ◡∅ = ∅
7	5, 6	eqtrdi 2788	. . . . . . . . 9 ⊢ (𝐹 = ∅ → ◡𝐹 = ∅)
8	2, 7	sylbi 217	. . . . . . . 8 ⊢ (𝐹 Fn ∅ → ◡𝐹 = ∅)
9	8	fneq1d 6593	. . . . . . 7 ⊢ (𝐹 Fn ∅ → (◡𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴))
10	9	biimpa 476	. . . . . 6 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → ∅ Fn 𝐴)
11	10	fndmd 6605	. . . . 5 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → dom ∅ = 𝐴)
12		dm0 5877	. . . . 5 ⊢ dom ∅ = ∅
13	11, 12	eqtr3di 2787	. . . 4 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → 𝐴 = ∅)
14	4, 13	jca 511	. . 3 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → (𝐹 = ∅ ∧ 𝐴 = ∅))
15	2	biimpri 228	. . . . 5 ⊢ (𝐹 = ∅ → 𝐹 Fn ∅)
16	15	adantr 480	. . . 4 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹 Fn ∅)
17		eqid 2737	. . . . . 6 ⊢ ∅ = ∅
18		fn0 6631	. . . . . 6 ⊢ (∅ Fn ∅ ↔ ∅ = ∅)
19	17, 18	mpbir 231	. . . . 5 ⊢ ∅ Fn ∅
20	7	fneq1d 6593	. . . . . 6 ⊢ (𝐹 = ∅ → (◡𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴))
21		fneq2 6592	. . . . . 6 ⊢ (𝐴 = ∅ → (∅ Fn 𝐴 ↔ ∅ Fn ∅))
22	20, 21	sylan9bb 509	. . . . 5 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → (◡𝐹 Fn 𝐴 ↔ ∅ Fn ∅))
23	19, 22	mpbiri 258	. . . 4 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → ◡𝐹 Fn 𝐴)
24	16, 23	jca 511	. . 3 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴))
25	14, 24	impbii 209	. 2 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
26	1, 25	bitri 275	1 ⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542  ∅c0 4287  ^◡ccnv 5631  dom cdm 5632   Fn wfn 6495  –1-1-onto→wf1o 6499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507

This theorem is referenced by:  fo00  6818  f1o0  6819  en0  8967  en0ALT  8968  en0r  8969  infn0  9214  symgbas0  19333  derang0  35389  poimirlem28  37903