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Mirrors > Home > ILE Home > Th. List > fvmptd | GIF version |
Description: Deduction version of fvmpt 5589. (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 5513 | . 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 3103 | . . . 4 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) |
6 | fvmptd.4 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
7 | 5, 6 | eqeltrd 2254 | . . 3 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝑉) |
8 | eqid 2177 | . . . 4 ⊢ (𝑥 ∈ 𝐷 ↦ 𝐵) = (𝑥 ∈ 𝐷 ↦ 𝐵) | |
9 | 8 | fvmpts 5590 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = ⦋𝐴 / 𝑥⦌𝐵) |
10 | 3, 7, 9 | syl2anc 411 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = ⦋𝐴 / 𝑥⦌𝐵) |
11 | 2, 10, 5 | 3eqtrd 2214 | 1 ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ⦋csb 3057 ↦ cmpt 4061 ‘cfv 5212 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-iota 5174 df-fun 5214 df-fv 5220 |
This theorem is referenced by: fvmptdv2 5601 rdgivallem 6376 1stinl 7067 2ndinl 7068 1stinr 7069 2ndinr 7070 updjudhcoinlf 7073 updjudhcoinrg 7074 cardcl 7174 caucvgsrlemfv 7778 caucvgsrlemoffval 7783 axcaucvglemval 7884 negiso 8898 infrenegsupex 9580 iseqf1olemfvp 10480 seq3f1olemqsum 10483 infxrnegsupex 11252 climcvg1nlem 11338 isumshft 11479 lmfval 13352 blfvalps 13545 cdivcncfap 13747 peano4nninf 14404 peano3nninf 14405 nninfsellemeq 14412 nninfsellemeqinf 14414 |
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