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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 6999 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 2909 | . . . 4 ⊢ Ⅎ𝑥 𝐶 ∈ V |
4 | fvmptf.4 | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) | |
5 | nfmpt1 5253 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐷 ↦ 𝐵) | |
6 | 4, 5 | nfcxfr 2890 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 |
7 | 6, 1 | nffv 6903 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝐴) |
8 | 7, 2 | nfeq 2906 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝐴) = 𝐶 |
9 | 3, 8 | nfim 1892 | . . 3 ⊢ Ⅎ𝑥(𝐶 ∈ V → (𝐹‘𝐴) = 𝐶) |
10 | fvmptf.3 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
11 | 10 | eleq1d 2811 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ V ↔ 𝐶 ∈ V)) |
12 | fveq2 6893 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴)) | |
13 | 12, 10 | eqeq12d 2742 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) = 𝐵 ↔ (𝐹‘𝐴) = 𝐶)) |
14 | 11, 13 | imbi12d 343 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ V → (𝐹‘𝑥) = 𝐵) ↔ (𝐶 ∈ V → (𝐹‘𝐴) = 𝐶))) |
15 | 4 | fvmpt2 7012 | . . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝐵 ∈ V) → (𝐹‘𝑥) = 𝐵) |
16 | 15 | ex 411 | . . 3 ⊢ (𝑥 ∈ 𝐷 → (𝐵 ∈ V → (𝐹‘𝑥) = 𝐵)) |
17 | 1, 9, 14, 16 | vtoclgaf 3556 | . 2 ⊢ (𝐴 ∈ 𝐷 → (𝐶 ∈ V → (𝐹‘𝐴) = 𝐶)) |
18 | elex 3482 | . 2 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V) | |
19 | 17, 18 | impel 504 | 1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘𝐴) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 Ⅎwnfc 2876 Vcvv 3462 ↦ cmpt 5228 ‘cfv 6546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fv 6554 |
This theorem is referenced by: fvmptnf 7023 elfvmptrab1w 7028 elfvmptrab1 7029 elovmpt3rab1 7678 rdgsucmptf 8450 frsucmpt 8460 fprodntriv 15939 prodss 15944 fprodefsum 16092 dvfsumabs 26045 dvfsumlem1 26048 dvfsumlem4 26052 dvfsum2 26057 dchrisumlem2 27516 dchrisumlem3 27517 rmfsupp2 33108 ptrest 37333 hlhilset 41646 fsumsermpt 45236 mulc1cncfg 45246 expcnfg 45248 climsubmpt 45317 climeldmeqmpt 45325 climfveqmpt 45328 fnlimfvre 45331 climfveqmpt3 45339 climeldmeqmpt3 45346 climinf2mpt 45371 climinfmpt 45372 stoweidlem23 45680 stoweidlem34 45691 stoweidlem36 45693 wallispilem5 45726 stirlinglem4 45734 stirlinglem11 45741 stirlinglem12 45742 stirlinglem13 45743 stirlinglem14 45744 sge0lempt 46067 sge0isummpt2 46089 meadjiun 46123 hoimbl2 46322 vonhoire 46329 |
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