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Mirrors > Home > NFE Home > Th. List > fvmpti | GIF version |
Description: Value of a function given in maps-to notation. (Contributed by Mario Carneiro, 23-Apr-2014.) |
Ref | Expression |
---|---|
fvmptg.1 | ⊢ (x = A → B = C) |
fvmptg.2 | ⊢ F = (x ∈ D ↦ B) |
Ref | Expression |
---|---|
fvmpti | ⊢ (A ∈ D → (F ‘A) = ( I ‘C)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvmptg.1 | . . . 4 ⊢ (x = A → B = C) | |
2 | fvmptg.2 | . . . 4 ⊢ F = (x ∈ D ↦ B) | |
3 | 1, 2 | fvmptg 5698 | . . 3 ⊢ ((A ∈ D ∧ C ∈ V) → (F ‘A) = C) |
4 | fvi 5442 | . . . 4 ⊢ (C ∈ V → ( I ‘C) = C) | |
5 | 4 | adantl 452 | . . 3 ⊢ ((A ∈ D ∧ C ∈ V) → ( I ‘C) = C) |
6 | 3, 5 | eqtr4d 2388 | . 2 ⊢ ((A ∈ D ∧ C ∈ V) → (F ‘A) = ( I ‘C)) |
7 | 1 | eleq1d 2419 | . . . . . . . 8 ⊢ (x = A → (B ∈ V ↔ C ∈ V)) |
8 | 2 | dmmpt 5683 | . . . . . . . 8 ⊢ dom F = {x ∈ D ∣ B ∈ V} |
9 | 7, 8 | elrab2 2996 | . . . . . . 7 ⊢ (A ∈ dom F ↔ (A ∈ D ∧ C ∈ V)) |
10 | 9 | baib 871 | . . . . . 6 ⊢ (A ∈ D → (A ∈ dom F ↔ C ∈ V)) |
11 | 10 | notbid 285 | . . . . 5 ⊢ (A ∈ D → (¬ A ∈ dom F ↔ ¬ C ∈ V)) |
12 | ndmfv 5349 | . . . . 5 ⊢ (¬ A ∈ dom F → (F ‘A) = ∅) | |
13 | 11, 12 | syl6bir 220 | . . . 4 ⊢ (A ∈ D → (¬ C ∈ V → (F ‘A) = ∅)) |
14 | 13 | imp 418 | . . 3 ⊢ ((A ∈ D ∧ ¬ C ∈ V) → (F ‘A) = ∅) |
15 | fvprc 5325 | . . . 4 ⊢ (¬ C ∈ V → ( I ‘C) = ∅) | |
16 | 15 | adantl 452 | . . 3 ⊢ ((A ∈ D ∧ ¬ C ∈ V) → ( I ‘C) = ∅) |
17 | 14, 16 | eqtr4d 2388 | . 2 ⊢ ((A ∈ D ∧ ¬ C ∈ V) → (F ‘A) = ( I ‘C)) |
18 | 6, 17 | pm2.61dan 766 | 1 ⊢ (A ∈ D → (F ‘A) = ( I ‘C)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 358 = wceq 1642 ∈ wcel 1710 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-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: fvmpt2i 5703 fvmptex 5721 |
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