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Mirrors > Home > MPE Home > Th. List > fvmpt2d | Structured version Visualization version GIF version |
Description: Deduction version of fvmpt2 6200. (Contributed by Thierry Arnoux, 8-Dec-2016.) |
Ref | Expression |
---|---|
fvmpt2d.1 | ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)) |
fvmpt2d.4 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) |
Ref | Expression |
---|---|
fvmpt2d | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvmpt2d.1 | . . . 4 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)) | |
2 | 1 | fveq1d 6105 | . . 3 ⊢ (𝜑 → (𝐹‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) |
3 | 2 | adantr 480 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) |
4 | id 22 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴) | |
5 | fvmpt2d.4 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) | |
6 | eqid 2610 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
7 | 6 | fvmpt2 6200 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵) |
8 | 4, 5, 7 | syl2an2 871 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵) |
9 | 3, 8 | eqtrd 2644 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ↦ cmpt 4643 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fv 5812 |
This theorem is referenced by: cantnflem1 8469 frlmphl 19939 neiptopreu 20747 rrxds 22989 ofoprabco 28847 esumcvg 29475 ofcfval2 29493 eulerpartgbij 29761 dstrvprob 29860 cvgdvgrat 37534 radcnvrat 37535 binomcxplemnotnn0 37577 fmuldfeqlem1 38649 cncficcgt0 38774 dvdivbd 38813 dvnmul 38833 dvnprodlem1 38836 dvnprodlem2 38837 stoweidlem42 38935 dirkeritg 38995 elaa2lem 39126 etransclem4 39131 ioorrnopnxrlem 39202 subsaliuncllem 39251 meaiuninclem 39373 meaiininclem 39376 ovnhoilem1 39491 ovncvr2 39501 ovolval4lem1 39539 iccvonmbllem 39569 vonioolem1 39571 vonioolem2 39572 vonicclem1 39574 vonicclem2 39575 pimconstlt0 39591 pimconstlt1 39592 pimgtmnf 39609 smfpimltmpt 39633 issmfdmpt 39635 smfpimltxrmpt 39645 smfaddlem2 39650 smflimlem2 39658 smflimlem4 39660 smfpimgtmpt 39667 smfpimgtxrmpt 39670 smfmullem4 39679 |
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