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Mirrors > Home > ILE Home > Th. List > fvmptd | GIF version |
Description: Deduction version of fvmpt 5270. (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 5200 | . 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 2948 | . . . 4 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) |
6 | fvmptd.4 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
7 | 5, 6 | eqeltrd 2155 | . . 3 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝑉) |
8 | eqid 2081 | . . . 4 ⊢ (𝑥 ∈ 𝐷 ↦ 𝐵) = (𝑥 ∈ 𝐷 ↦ 𝐵) | |
9 | 8 | fvmpts 5271 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = ⦋𝐴 / 𝑥⦌𝐵) |
10 | 3, 7, 9 | syl2anc 403 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = ⦋𝐴 / 𝑥⦌𝐵) |
11 | 2, 10, 5 | 3eqtrd 2117 | 1 ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ∈ wcel 1433 ⦋csb 2908 ↦ cmpt 3839 ‘cfv 4922 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-csb 2909 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 |
This theorem is referenced by: fvmptdv2 5281 rdgivallem 5991 cardcl 6450 caucvgsrlemfv 6967 caucvgsrlemoffval 6972 axcaucvglemval 7063 negiso 8033 infrenegsupex 8682 climcvg1nlem 10186 |
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