|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > fvmptf | Structured version Visualization version GIF version | ||
| Description: Value of a function given by an ordered-pair class abstraction. This version of fvmptg 7013 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.) | 
| Ref | Expression | 
|---|---|
| fvmptf.1 | ⊢ Ⅎ𝑥𝐴 | 
| fvmptf.2 | ⊢ Ⅎ𝑥𝐶 | 
| fvmptf.3 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | 
| fvmptf.4 | ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) | 
| Ref | Expression | 
|---|---|
| fvmptf | ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘𝐴) = 𝐶) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fvmptf.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
| 2 | fvmptf.2 | . . . . 5 ⊢ Ⅎ𝑥𝐶 | |
| 3 | 2 | nfel1 2921 | . . . 4 ⊢ Ⅎ𝑥 𝐶 ∈ V | 
| 4 | fvmptf.4 | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) | |
| 5 | nfmpt1 5249 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐷 ↦ 𝐵) | |
| 6 | 4, 5 | nfcxfr 2902 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 | 
| 7 | 6, 1 | nffv 6915 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝐴) | 
| 8 | 7, 2 | nfeq 2918 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝐴) = 𝐶 | 
| 9 | 3, 8 | nfim 1895 | . . 3 ⊢ Ⅎ𝑥(𝐶 ∈ V → (𝐹‘𝐴) = 𝐶) | 
| 10 | fvmptf.3 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
| 11 | 10 | eleq1d 2825 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ V ↔ 𝐶 ∈ V)) | 
| 12 | fveq2 6905 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴)) | |
| 13 | 12, 10 | eqeq12d 2752 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) = 𝐵 ↔ (𝐹‘𝐴) = 𝐶)) | 
| 14 | 11, 13 | imbi12d 344 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ V → (𝐹‘𝑥) = 𝐵) ↔ (𝐶 ∈ V → (𝐹‘𝐴) = 𝐶))) | 
| 15 | 4 | fvmpt2 7026 | . . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝐵 ∈ V) → (𝐹‘𝑥) = 𝐵) | 
| 16 | 15 | ex 412 | . . 3 ⊢ (𝑥 ∈ 𝐷 → (𝐵 ∈ V → (𝐹‘𝑥) = 𝐵)) | 
| 17 | 1, 9, 14, 16 | vtoclgaf 3575 | . 2 ⊢ (𝐴 ∈ 𝐷 → (𝐶 ∈ V → (𝐹‘𝐴) = 𝐶)) | 
| 18 | elex 3500 | . 2 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V) | |
| 19 | 17, 18 | impel 505 | 1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘𝐴) = 𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Ⅎwnfc 2889 Vcvv 3479 ↦ cmpt 5224 ‘cfv 6560 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fv 6568 | 
| This theorem is referenced by: fvmptnf 7037 elfvmptrab1w 7042 elfvmptrab1 7043 elovmpt3rab1 7694 rdgsucmptf 8469 frsucmpt 8479 fprodntriv 15979 prodss 15984 fprodefsum 16132 dvfsumabs 26064 dvfsumlem1 26067 dvfsumlem4 26071 dvfsum2 26076 dchrisumlem2 27535 dchrisumlem3 27536 rmfsupp2 33243 ptrest 37627 hlhilset 41937 fsumsermpt 45599 mulc1cncfg 45609 expcnfg 45611 climsubmpt 45680 climeldmeqmpt 45688 climfveqmpt 45691 fnlimfvre 45694 climfveqmpt3 45702 climeldmeqmpt3 45709 climinf2mpt 45734 climinfmpt 45735 stoweidlem23 46043 stoweidlem34 46054 stoweidlem36 46056 wallispilem5 46089 stirlinglem4 46097 stirlinglem11 46104 stirlinglem12 46105 stirlinglem13 46106 stirlinglem14 46107 sge0lempt 46430 sge0isummpt2 46452 meadjiun 46486 hoimbl2 46685 vonhoire 46692 | 
| Copyright terms: Public domain | W3C validator |