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 6385 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 1886 | . . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) → [𝑦 / 𝑥]𝐵 ∈ 𝐶) |
3 | sban 2399 | . . . . . 6 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝑥 ∈ 𝐴)) | |
4 | fmptdF.p | . . . . . . . 8 ⊢ Ⅎ𝑥𝜑 | |
5 | 4 | sbf 2380 | . . . . . . 7 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑) |
6 | fmptdF.a | . . . . . . . 8 ⊢ Ⅎ𝑥𝐴 | |
7 | 6 | clelsb3f 2768 | . . . . . . 7 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) |
8 | 5, 7 | anbi12i 733 | . . . . . 6 ⊢ (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑦 ∈ 𝐴)) |
9 | 3, 8 | bitri 264 | . . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑦 ∈ 𝐴)) |
10 | sbsbc 3439 | . . . . . 6 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ [𝑦 / 𝑥]𝐵 ∈ 𝐶) | |
11 | sbcel12 3983 | . . . . . . 7 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ ⦋𝑦 / 𝑥⦌𝐶) | |
12 | vex 3203 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
13 | fmptdF.c | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝐶 | |
14 | 13 | csbconstgf 3545 | . . . . . . . . 9 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑥⦌𝐶 = 𝐶) |
15 | 12, 14 | ax-mp 5 | . . . . . . . 8 ⊢ ⦋𝑦 / 𝑥⦌𝐶 = 𝐶 |
16 | 15 | eleq2i 2693 | . . . . . . 7 ⊢ (⦋𝑦 / 𝑥⦌𝐵 ∈ ⦋𝑦 / 𝑥⦌𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶) |
17 | 11, 16 | bitri 264 | . . . . . 6 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶) |
18 | 10, 17 | bitri 264 | . . . . 5 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶) |
19 | 2, 9, 18 | 3imtr3i 280 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶) |
20 | 19 | ralrimiva 2966 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶) |
21 | nfcv 2764 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
22 | nfcv 2764 | . . . . 5 ⊢ Ⅎ𝑦𝐵 | |
23 | nfcsb1v 3549 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
24 | csbeq1a 3542 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
25 | 6, 21, 22, 23, 24 | cbvmptf 4748 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵) |
26 | 25 | fmpt 6381 | . . 3 ⊢ (∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶) |
27 | 20, 26 | sylib 208 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶) |
28 | fmptdF.2 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
29 | 28 | feq1i 6036 | . 2 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶) |
30 | 27, 29 | sylibr 224 | 1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 Ⅎwnf 1708 [wsb 1880 ∈ wcel 1990 Ⅎwnfc 2751 ∀wral 2912 Vcvv 3200 [wsbc 3435 ⦋csb 3533 ↦ cmpt 4729 ⟶wf 5884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 |
This theorem is referenced by: fmptcof2 29457 esumcl 30092 esumid 30106 esumgsum 30107 esumval 30108 esumel 30109 esumsplit 30115 esumaddf 30123 esumss 30134 esumpfinvalf 30138 |
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