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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 6760 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 2994 | . . . 4 ⊢ Ⅎ𝑥 𝐶 ∈ V |
4 | fvmptf.4 | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) | |
5 | nfmpt1 5156 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐷 ↦ 𝐵) | |
6 | 4, 5 | nfcxfr 2975 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 |
7 | 6, 1 | nffv 6674 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝐴) |
8 | 7, 2 | nfeq 2991 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝐴) = 𝐶 |
9 | 3, 8 | nfim 1893 | . . 3 ⊢ Ⅎ𝑥(𝐶 ∈ V → (𝐹‘𝐴) = 𝐶) |
10 | fvmptf.3 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
11 | 10 | eleq1d 2897 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ V ↔ 𝐶 ∈ V)) |
12 | fveq2 6664 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴)) | |
13 | 12, 10 | eqeq12d 2837 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) = 𝐵 ↔ (𝐹‘𝐴) = 𝐶)) |
14 | 11, 13 | imbi12d 347 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ V → (𝐹‘𝑥) = 𝐵) ↔ (𝐶 ∈ V → (𝐹‘𝐴) = 𝐶))) |
15 | 4 | fvmpt2 6773 | . . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝐵 ∈ V) → (𝐹‘𝑥) = 𝐵) |
16 | 15 | ex 415 | . . 3 ⊢ (𝑥 ∈ 𝐷 → (𝐵 ∈ V → (𝐹‘𝑥) = 𝐵)) |
17 | 1, 9, 14, 16 | vtoclgaf 3572 | . 2 ⊢ (𝐴 ∈ 𝐷 → (𝐶 ∈ V → (𝐹‘𝐴) = 𝐶)) |
18 | elex 3512 | . 2 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V) | |
19 | 17, 18 | impel 508 | 1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘𝐴) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Ⅎwnfc 2961 Vcvv 3494 ↦ cmpt 5138 ‘cfv 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fv 6357 |
This theorem is referenced by: fvmptnf 6784 elfvmptrab1w 6788 elfvmptrab1 6789 elovmpt3rab1 7399 rdgsucmptf 8058 frsucmpt 8067 fprodntriv 15290 prodss 15295 fprodefsum 15442 dvfsumabs 24614 dvfsumlem1 24617 dvfsumlem4 24620 dvfsum2 24625 dchrisumlem2 26060 dchrisumlem3 26061 rmfsupp2 30861 ptrest 34885 hlhilset 39064 fsumsermpt 41853 mulc1cncfg 41863 expcnfg 41865 climsubmpt 41934 climeldmeqmpt 41942 climfveqmpt 41945 fnlimfvre 41948 climfveqmpt3 41956 climeldmeqmpt3 41963 climinf2mpt 41988 climinfmpt 41989 stoweidlem23 42302 stoweidlem34 42313 stoweidlem36 42315 wallispilem5 42348 stirlinglem4 42356 stirlinglem11 42363 stirlinglem12 42364 stirlinglem13 42365 stirlinglem14 42366 sge0lempt 42686 sge0isummpt2 42708 meadjiun 42742 hoimbl2 42941 vonhoire 42948 |
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