Theorem f1o00 6856

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 6829	. 2 ⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴))
2		fn0 6666	. . . . 5 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅)
3	2	birani 509	. . . 4 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → 𝐹 = ∅)
4		cnveq 5855	. . . . . . . . . 10 ⊢ (𝐹 = ∅ → ◡𝐹 = ◡∅)
5		cnv0 5865	. . . . . . . . . 10 ⊢ ◡∅ = ∅
6	4, 5	eqtrdi 2811	. . . . . . . . 9 ⊢ (𝐹 = ∅ → ◡𝐹 = ∅)
7	2, 6	sylbi 220	. . . . . . . 8 ⊢ (𝐹 Fn ∅ → ◡𝐹 = ∅)
8	7	fneq1d 6628	. . . . . . 7 ⊢ (𝐹 Fn ∅ → (◡𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴))
9	8	biimpa 482	. . . . . 6 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → ∅ Fn 𝐴)
10	9	fndmd 6640	. . . . 5 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → dom ∅ = 𝐴)
11		dm0 5906	. . . . 5 ⊢ dom ∅ = ∅
12	10, 11	eqtr3di 2810	. . . 4 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → 𝐴 = ∅)
13	3, 12	jca 521	. . 3 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → (𝐹 = ∅ ∧ 𝐴 = ∅))
14	2	biranri 511	. . . 4 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹 Fn ∅)
15		eqid 2760	. . . . . 6 ⊢ ∅ = ∅
16		fn0 6666	. . . . . 6 ⊢ (∅ Fn ∅ ↔ ∅ = ∅)
17	15, 16	mpbir 234	. . . . 5 ⊢ ∅ Fn ∅
18	6	fneq1d 6628	. . . . . 6 ⊢ (𝐹 = ∅ → (◡𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴))
19		fneq2 6627	. . . . . 6 ⊢ (𝐴 = ∅ → (∅ Fn 𝐴 ↔ ∅ Fn ∅))
20	18, 19	sylan9bb 519	. . . . 5 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → (◡𝐹 Fn 𝐴 ↔ ∅ Fn ∅))
21	17, 20	mpbiri 261	. . . 4 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → ◡𝐹 Fn 𝐴)
22	14, 21	jca 521	. . 3 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴))
23	13, 22	impbii 212	. 2 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
24	1, 23	bitri 278	1 ⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∅c0 4279  ^◡ccnv 5654  dom cdm 5655   Fn wfn 6530  –1-1-onto→wf1o 6534

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542

This theorem is used by:  fo00  6857  f1o0  6858  en0  9031  en0ALT  9032  en0r  9033  infn0  9279  symgbas0  19542  derang0  35831  poimirlem28  38462