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 co-domain of the mapping operation; deduction form. This version of fmptd 6292 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 1873 | . . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) → [𝑦 / 𝑥]𝐵 ∈ 𝐶) |
3 | sban 2387 | . . . . . 6 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝑥 ∈ 𝐴)) | |
4 | fmptdF.p | . . . . . . . 8 ⊢ Ⅎ𝑥𝜑 | |
5 | 4 | sbf 2368 | . . . . . . 7 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑) |
6 | fmptdF.a | . . . . . . . 8 ⊢ Ⅎ𝑥𝐴 | |
7 | 6 | clelsb3f 28704 | . . . . . . 7 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) |
8 | 5, 7 | anbi12i 729 | . . . . . 6 ⊢ (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑦 ∈ 𝐴)) |
9 | 3, 8 | bitri 263 | . . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑦 ∈ 𝐴)) |
10 | sbsbc 3406 | . . . . . 6 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ [𝑦 / 𝑥]𝐵 ∈ 𝐶) | |
11 | sbcel12 3935 | . . . . . . 7 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ ⦋𝑦 / 𝑥⦌𝐶) | |
12 | vex 3176 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
13 | fmptdF.c | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝐶 | |
14 | 13 | csbconstgf 3511 | . . . . . . . . 9 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑥⦌𝐶 = 𝐶) |
15 | 12, 14 | ax-mp 5 | . . . . . . . 8 ⊢ ⦋𝑦 / 𝑥⦌𝐶 = 𝐶 |
16 | 15 | eleq2i 2680 | . . . . . . 7 ⊢ (⦋𝑦 / 𝑥⦌𝐵 ∈ ⦋𝑦 / 𝑥⦌𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶) |
17 | 11, 16 | bitri 263 | . . . . . 6 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶) |
18 | 10, 17 | bitri 263 | . . . . 5 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶) |
19 | 2, 9, 18 | 3imtr3i 279 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶) |
20 | 19 | ralrimiva 2949 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶) |
21 | nfcv 2751 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
22 | nfcv 2751 | . . . . 5 ⊢ Ⅎ𝑦𝐵 | |
23 | nfcsb1v 3515 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
24 | csbeq1a 3508 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
25 | 6, 21, 22, 23, 24 | cbvmptf 4676 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵) |
26 | 25 | fmpt 6289 | . . 3 ⊢ (∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶) |
27 | 20, 26 | sylib 207 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶) |
28 | fmptdF.2 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
29 | 28 | feq1i 5949 | . 2 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶) |
30 | 27, 29 | sylibr 223 | 1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 Ⅎwnf 1699 [wsb 1867 ∈ wcel 1977 Ⅎwnfc 2738 ∀wral 2896 Vcvv 3173 [wsbc 3402 ⦋csb 3499 ↦ cmpt 4643 ⟶wf 5800 |
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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-fal 1481 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-ne 2782 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-fn 5807 df-f 5808 df-fv 5812 |
This theorem is referenced by: fmptcof2 28839 esumcl 29419 esumid 29433 esumgsum 29434 esumval 29435 esumel 29436 esumsplit 29442 esumaddf 29450 esumss 29461 esumpfinvalf 29465 |
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