Theorem f1o00 5317

Description: One-to-one onto mapping of the empty set. (Contributed by set.mm contributors, 15-Apr-1998.)

Assertion

Ref	Expression
f1o00	⊢ (F:∅–1-1-onto→A ↔ (F = ∅ ∧ A = ∅))

Proof of Theorem f1o00

Step	Hyp	Ref	Expression
1		dff1o4 5294	. 2 ⊢ (F:∅–1-1-onto→A ↔ (F Fn ∅ ∧ ◡F Fn A))
2		fn0 5202	. . . . . 6 ⊢ (F Fn ∅ ↔ F = ∅)
3	2	biimpi 186	. . . . 5 ⊢ (F Fn ∅ → F = ∅)
4	3	adantr 451	. . . 4 ⊢ ((F Fn ∅ ∧ ◡F Fn A) → F = ∅)
5		dm0 4918	. . . . 5 ⊢ dom ∅ = ∅
6		cnveq 4886	. . . . . . . . . 10 ⊢ (F = ∅ → ◡F = ◡∅)
7		cnv0 5031	. . . . . . . . . 10 ⊢ ◡∅ = ∅
8	6, 7	syl6eq 2401	. . . . . . . . 9 ⊢ (F = ∅ → ◡F = ∅)
9	2, 8	sylbi 187	. . . . . . . 8 ⊢ (F Fn ∅ → ◡F = ∅)
10	9	fneq1d 5175	. . . . . . 7 ⊢ (F Fn ∅ → (◡F Fn A ↔ ∅ Fn A))
11	10	biimpa 470	. . . . . 6 ⊢ ((F Fn ∅ ∧ ◡F Fn A) → ∅ Fn A)
12		fndm 5182	. . . . . 6 ⊢ (∅ Fn A → dom ∅ = A)
13	11, 12	syl 15	. . . . 5 ⊢ ((F Fn ∅ ∧ ◡F Fn A) → dom ∅ = A)
14	5, 13	syl5reqr 2400	. . . 4 ⊢ ((F Fn ∅ ∧ ◡F Fn A) → A = ∅)
15	4, 14	jca 518	. . 3 ⊢ ((F Fn ∅ ∧ ◡F Fn A) → (F = ∅ ∧ A = ∅))
16	2	biimpri 197	. . . . 5 ⊢ (F = ∅ → F Fn ∅)
17	16	adantr 451	. . . 4 ⊢ ((F = ∅ ∧ A = ∅) → F Fn ∅)
18		eqid 2353	. . . . . 6 ⊢ ∅ = ∅
19		fn0 5202	. . . . . 6 ⊢ (∅ Fn ∅ ↔ ∅ = ∅)
20	18, 19	mpbir 200	. . . . 5 ⊢ ∅ Fn ∅
21	8	fneq1d 5175	. . . . . 6 ⊢ (F = ∅ → (◡F Fn A ↔ ∅ Fn A))
22		fneq2 5174	. . . . . 6 ⊢ (A = ∅ → (∅ Fn A ↔ ∅ Fn ∅))
23	21, 22	sylan9bb 680	. . . . 5 ⊢ ((F = ∅ ∧ A = ∅) → (◡F Fn A ↔ ∅ Fn ∅))
24	20, 23	mpbiri 224	. . . 4 ⊢ ((F = ∅ ∧ A = ∅) → ◡F Fn A)
25	17, 24	jca 518	. . 3 ⊢ ((F = ∅ ∧ A = ∅) → (F Fn ∅ ∧ ◡F Fn A))
26	15, 25	impbii 180	. 2 ⊢ ((F Fn ∅ ∧ ◡F Fn A) ↔ (F = ∅ ∧ A = ∅))
27	1, 26	bitri 240	1 ⊢ (F:∅–1-1-onto→A ↔ (F = ∅ ∧ A = ∅))

Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642  ∅c0 3550  ^◡ccnv 4771  dom cdm 4772   Fn wfn 4776  –1-1-onto→wf1o 4780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794

This theorem is referenced by:  fo00  5318  en0  6042