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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmptdF | Structured version Visualization version GIF version |
Description: Domain and codomain of the mapping operation; deduction form. This version of fmptd 7098 uses bound-variable hypothesis instead of distinct variable conditions. (Contributed by Thierry Arnoux, 28-Mar-2017.) |
Ref | Expression |
---|---|
fmptdF.p | ⊢ Ⅎ𝑥𝜑 |
fmptdF.a | ⊢ Ⅎ𝑥𝐴 |
fmptdF.c | ⊢ Ⅎ𝑥𝐶 |
fmptdF.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) |
fmptdF.2 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
Ref | Expression |
---|---|
fmptdF | ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fmptdF.1 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) | |
2 | 1 | sbimi 2077 | . . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) → [𝑦 / 𝑥]𝐵 ∈ 𝐶) |
3 | sban 2083 | . . . . . 6 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝑥 ∈ 𝐴)) | |
4 | fmptdF.p | . . . . . . . 8 ⊢ Ⅎ𝑥𝜑 | |
5 | 4 | sbf 2262 | . . . . . . 7 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑) |
6 | fmptdF.a | . . . . . . . 8 ⊢ Ⅎ𝑥𝐴 | |
7 | 6 | clelsb1fw 2906 | . . . . . . 7 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) |
8 | 5, 7 | anbi12i 627 | . . . . . 6 ⊢ (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑦 ∈ 𝐴)) |
9 | 3, 8 | bitri 274 | . . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑦 ∈ 𝐴)) |
10 | sbsbc 3777 | . . . . . 6 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ [𝑦 / 𝑥]𝐵 ∈ 𝐶) | |
11 | sbcel12 4404 | . . . . . . 7 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ ⦋𝑦 / 𝑥⦌𝐶) | |
12 | vex 3477 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
13 | fmptdF.c | . . . . . . . . 9 ⊢ Ⅎ𝑥𝐶 | |
14 | 12, 13 | csbgfi 3910 | . . . . . . . 8 ⊢ ⦋𝑦 / 𝑥⦌𝐶 = 𝐶 |
15 | 14 | eleq2i 2824 | . . . . . . 7 ⊢ (⦋𝑦 / 𝑥⦌𝐵 ∈ ⦋𝑦 / 𝑥⦌𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶) |
16 | 11, 15 | bitri 274 | . . . . . 6 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶) |
17 | 10, 16 | bitri 274 | . . . . 5 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶) |
18 | 2, 9, 17 | 3imtr3i 290 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶) |
19 | 18 | ralrimiva 3145 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶) |
20 | nfcv 2902 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
21 | nfcv 2902 | . . . . 5 ⊢ Ⅎ𝑦𝐵 | |
22 | nfcsb1v 3914 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
23 | csbeq1a 3903 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
24 | 6, 20, 21, 22, 23 | cbvmptf 5250 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵) |
25 | 24 | fmpt 7094 | . . 3 ⊢ (∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶) |
26 | 19, 25 | sylib 217 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶) |
27 | fmptdF.2 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
28 | 27 | feq1i 6695 | . 2 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶) |
29 | 26, 28 | sylibr 233 | 1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 Ⅎwnf 1785 [wsb 2067 ∈ wcel 2106 Ⅎwnfc 2882 ∀wral 3060 [wsbc 3773 ⦋csb 3889 ↦ cmpt 5224 ⟶wf 6528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-fun 6534 df-fn 6535 df-f 6536 |
This theorem is referenced by: fmptcof2 31751 esumcl 32859 esumid 32873 esumgsum 32874 esumval 32875 esumel 32876 esumsplit 32882 esumaddf 32890 esumss 32901 esumpfinvalf 32905 |
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