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Mirrors > Home > NFE Home > Th. List > nfunsn | GIF version |
Description: If the restriction of a class to a singleton is not a function, its value is the empty set. (Contributed by NM, 8-Aug-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
nfunsn | ⊢ (¬ Fun (F ↾ {A}) → (F ‘A) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eumo 2244 | . . . . . 6 ⊢ (∃!y AFy → ∃*y AFy) | |
2 | brres 4949 | . . . . . . . 8 ⊢ (x(F ↾ {A})y ↔ (xFy ∧ x ∈ {A})) | |
3 | elsn 3748 | . . . . . . . . . 10 ⊢ (x ∈ {A} ↔ x = A) | |
4 | breq1 4642 | . . . . . . . . . 10 ⊢ (x = A → (xFy ↔ AFy)) | |
5 | 3, 4 | sylbi 187 | . . . . . . . . 9 ⊢ (x ∈ {A} → (xFy ↔ AFy)) |
6 | 5 | biimpac 472 | . . . . . . . 8 ⊢ ((xFy ∧ x ∈ {A}) → AFy) |
7 | 2, 6 | sylbi 187 | . . . . . . 7 ⊢ (x(F ↾ {A})y → AFy) |
8 | 7 | moimi 2251 | . . . . . 6 ⊢ (∃*y AFy → ∃*y x(F ↾ {A})y) |
9 | 1, 8 | syl 15 | . . . . 5 ⊢ (∃!y AFy → ∃*y x(F ↾ {A})y) |
10 | tz6.12-2 5346 | . . . . 5 ⊢ (¬ ∃!y AFy → (F ‘A) = ∅) | |
11 | 9, 10 | nsyl4 134 | . . . 4 ⊢ (¬ (F ‘A) = ∅ → ∃*y x(F ↾ {A})y) |
12 | 11 | alrimiv 1631 | . . 3 ⊢ (¬ (F ‘A) = ∅ → ∀x∃*y x(F ↾ {A})y) |
13 | dffun6 5124 | . . 3 ⊢ (Fun (F ↾ {A}) ↔ ∀x∃*y x(F ↾ {A})y) | |
14 | 12, 13 | sylibr 203 | . 2 ⊢ (¬ (F ‘A) = ∅ → Fun (F ↾ {A})) |
15 | 14 | con1i 121 | 1 ⊢ (¬ Fun (F ↾ {A}) → (F ‘A) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 176 ∧ wa 358 ∀wal 1540 = wceq 1642 ∈ wcel 1710 ∃!weu 2204 ∃*wmo 2205 ∅c0 3550 {csn 3737 class class class wbr 4639 ↾ cres 4774 Fun wfun 4775 ‘cfv 4781 |
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-id 4767 df-xp 4784 df-cnv 4785 df-res 4788 df-fun 4789 df-fv 4795 |
This theorem is referenced by: (None) |
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