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Mirrors > Home > ILE Home > Th. List > fvmptd | GIF version |
Description: Deduction version of fvmpt 5249. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
fvmptd.1 | ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)) |
fvmptd.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶) |
fvmptd.3 | ⊢ (𝜑 → 𝐴 ∈ 𝐷) |
fvmptd.4 | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
Ref | Expression |
---|---|
fvmptd | ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvmptd.1 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)) | |
2 | 1 | fveq1d 5180 | . 2 ⊢ (𝜑 → (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴)) |
3 | fvmptd.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐷) | |
4 | fvmptd.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶) | |
5 | 3, 4 | csbied 2892 | . . . 4 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) |
6 | fvmptd.4 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
7 | 5, 6 | eqeltrd 2114 | . . 3 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝑉) |
8 | eqid 2040 | . . . 4 ⊢ (𝑥 ∈ 𝐷 ↦ 𝐵) = (𝑥 ∈ 𝐷 ↦ 𝐵) | |
9 | 8 | fvmpts 5250 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = ⦋𝐴 / 𝑥⦌𝐵) |
10 | 3, 7, 9 | syl2anc 391 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = ⦋𝐴 / 𝑥⦌𝐵) |
11 | 2, 10, 5 | 3eqtrd 2076 | 1 ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 ⦋csb 2852 ↦ cmpt 3818 ‘cfv 4902 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fv 4910 |
This theorem is referenced by: fvmptdv2 5260 rdgivallem 5968 cardcl 6361 caucvgsrlemfv 6875 caucvgsrlemoffval 6880 axcaucvglemval 6971 climcvg1nlem 9868 |
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