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Mirrors > Home > NFE Home > Th. List > fvmptnf | GIF version |
Description: The value of a function given by an ordered-pair class abstraction is the empty set when the class it would otherwise map to is a proper class. This version of fvmptn 5724 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 11-Sep-2015.) |
Ref | Expression |
---|---|
fvmptf.1 | ⊢ ℲxA |
fvmptf.2 | ⊢ ℲxC |
fvmptf.3 | ⊢ (x = A → B = C) |
fvmptf.4 | ⊢ F = (x ∈ D ↦ B) |
Ref | Expression |
---|---|
fvmptnf | ⊢ (¬ C ∈ V → (F ‘A) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvmptf.4 | . . . . 5 ⊢ F = (x ∈ D ↦ B) | |
2 | 1 | dmmptss 5685 | . . . 4 ⊢ dom F ⊆ D |
3 | 2 | sseli 3269 | . . 3 ⊢ (A ∈ dom F → A ∈ D) |
4 | eqid 2353 | . . . . . . 7 ⊢ (x ∈ D ↦ ( I ‘B)) = (x ∈ D ↦ ( I ‘B)) | |
5 | 1, 4 | fvmptex 5721 | . . . . . 6 ⊢ (F ‘A) = ((x ∈ D ↦ ( I ‘B)) ‘A) |
6 | fvex 5339 | . . . . . . 7 ⊢ ( I ‘C) ∈ V | |
7 | fvmptf.1 | . . . . . . . 8 ⊢ ℲxA | |
8 | nfcv 2489 | . . . . . . . . 9 ⊢ Ⅎx I | |
9 | fvmptf.2 | . . . . . . . . 9 ⊢ ℲxC | |
10 | 8, 9 | nffv 5334 | . . . . . . . 8 ⊢ Ⅎx( I ‘C) |
11 | fvmptf.3 | . . . . . . . . 9 ⊢ (x = A → B = C) | |
12 | 11 | fveq2d 5332 | . . . . . . . 8 ⊢ (x = A → ( I ‘B) = ( I ‘C)) |
13 | 7, 10, 12, 4 | fvmptf 5722 | . . . . . . 7 ⊢ ((A ∈ D ∧ ( I ‘C) ∈ V) → ((x ∈ D ↦ ( I ‘B)) ‘A) = ( I ‘C)) |
14 | 6, 13 | mpan2 652 | . . . . . 6 ⊢ (A ∈ D → ((x ∈ D ↦ ( I ‘B)) ‘A) = ( I ‘C)) |
15 | 5, 14 | syl5eq 2397 | . . . . 5 ⊢ (A ∈ D → (F ‘A) = ( I ‘C)) |
16 | fvprc 5325 | . . . . 5 ⊢ (¬ C ∈ V → ( I ‘C) = ∅) | |
17 | 15, 16 | sylan9eq 2405 | . . . 4 ⊢ ((A ∈ D ∧ ¬ C ∈ V) → (F ‘A) = ∅) |
18 | 17 | expcom 424 | . . 3 ⊢ (¬ C ∈ V → (A ∈ D → (F ‘A) = ∅)) |
19 | 3, 18 | syl5 28 | . 2 ⊢ (¬ C ∈ V → (A ∈ dom F → (F ‘A) = ∅)) |
20 | ndmfv 5349 | . 2 ⊢ (¬ A ∈ dom F → (F ‘A) = ∅) | |
21 | 19, 20 | pm2.61d1 151 | 1 ⊢ (¬ C ∈ V → (F ‘A) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1642 ∈ wcel 1710 Ⅎwnfc 2476 Vcvv 2859 ∅c0 3550 I cid 4763 dom cdm 4772 ‘cfv 4781 ↦ cmpt 5651 |
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-csb 3137 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-id 4767 df-cnv 4785 df-rn 4786 df-dm 4787 df-fun 4789 df-fn 4790 df-fv 4795 df-mpt 5652 |
This theorem is referenced by: fvmptn 5724 |
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