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Mirrors > Home > NFE Home > Th. List > f1o00 | GIF version |
Description: One-to-one onto mapping of the empty set. (Contributed by set.mm contributors, 15-Apr-1998.) |
Ref | Expression |
---|---|
f1o00 | ⊢ (F:∅–1-1-onto→A ↔ (F = ∅ ∧ A = ∅)) |
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 = ∅)) |
Colors of variables: wff setvar class |
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-1 5 ax-2 6 ax-3 7 ax-mp 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 |
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