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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 6768 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 2996 | . . . 4 ⊢ Ⅎ𝑥 𝐶 ∈ V |
4 | fvmptf.4 | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) | |
5 | nfmpt1 5166 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐷 ↦ 𝐵) | |
6 | 4, 5 | nfcxfr 2977 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 |
7 | 6, 1 | nffv 6682 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝐴) |
8 | 7, 2 | nfeq 2993 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝐴) = 𝐶 |
9 | 3, 8 | nfim 1897 | . . 3 ⊢ Ⅎ𝑥(𝐶 ∈ V → (𝐹‘𝐴) = 𝐶) |
10 | fvmptf.3 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
11 | 10 | eleq1d 2899 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝐵 ∈ V ↔ 𝐶 ∈ V)) |
12 | fveq2 6672 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴)) | |
13 | 12, 10 | eqeq12d 2839 | . . . 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 6781 | . . . 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 3575 | . 2 ⊢ (𝐴 ∈ 𝐷 → (𝐶 ∈ V → (𝐹‘𝐴) = 𝐶)) |
18 | elex 3514 | . 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 1537 ∈ wcel 2114 Ⅎwnfc 2963 Vcvv 3496 ↦ cmpt 5148 ‘cfv 6357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fv 6365 |
This theorem is referenced by: fvmptnf 6792 elfvmptrab1w 6796 elfvmptrab1 6797 elovmpt3rab1 7407 rdgsucmptf 8066 frsucmpt 8075 fprodntriv 15298 prodss 15303 fprodefsum 15450 dvfsumabs 24622 dvfsumlem1 24625 dvfsumlem4 24628 dvfsum2 24633 dchrisumlem2 26068 dchrisumlem3 26069 rmfsupp2 30868 ptrest 34893 hlhilset 39072 fsumsermpt 41867 mulc1cncfg 41877 expcnfg 41879 climsubmpt 41948 climeldmeqmpt 41956 climfveqmpt 41959 fnlimfvre 41962 climfveqmpt3 41970 climeldmeqmpt3 41977 climinf2mpt 42002 climinfmpt 42003 stoweidlem23 42315 stoweidlem34 42326 stoweidlem36 42328 wallispilem5 42361 stirlinglem4 42369 stirlinglem11 42376 stirlinglem12 42377 stirlinglem13 42378 stirlinglem14 42379 sge0lempt 42699 sge0isummpt2 42721 meadjiun 42755 hoimbl2 42954 vonhoire 42961 |
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