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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 7134 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 2074 | . . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) → [𝑦 / 𝑥]𝐵 ∈ 𝐶) | 
| 3 | sban 2080 | . . . . . 6 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝑥 ∈ 𝐴)) | |
| 4 | fmptdF.p | . . . . . . . 8 ⊢ Ⅎ𝑥𝜑 | |
| 5 | 4 | sbf 2271 | . . . . . . 7 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑) | 
| 6 | fmptdF.a | . . . . . . . 8 ⊢ Ⅎ𝑥𝐴 | |
| 7 | 6 | clelsb1fw 2909 | . . . . . . 7 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) | 
| 8 | 5, 7 | anbi12i 628 | . . . . . 6 ⊢ (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑦 ∈ 𝐴)) | 
| 9 | 3, 8 | bitri 275 | . . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑦 ∈ 𝐴)) | 
| 10 | sbsbc 3792 | . . . . . 6 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ [𝑦 / 𝑥]𝐵 ∈ 𝐶) | |
| 11 | sbcel12 4411 | . . . . . . 7 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ ⦋𝑦 / 𝑥⦌𝐶) | |
| 12 | vex 3484 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
| 13 | fmptdF.c | . . . . . . . . 9 ⊢ Ⅎ𝑥𝐶 | |
| 14 | 12, 13 | csbgfi 3919 | . . . . . . . 8 ⊢ ⦋𝑦 / 𝑥⦌𝐶 = 𝐶 | 
| 15 | 14 | eleq2i 2833 | . . . . . . 7 ⊢ (⦋𝑦 / 𝑥⦌𝐵 ∈ ⦋𝑦 / 𝑥⦌𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶) | 
| 16 | 11, 15 | bitri 275 | . . . . . 6 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶) | 
| 17 | 10, 16 | bitri 275 | . . . . 5 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶) | 
| 18 | 2, 9, 17 | 3imtr3i 291 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶) | 
| 19 | 18 | ralrimiva 3146 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶) | 
| 20 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
| 21 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑦𝐵 | |
| 22 | nfcsb1v 3923 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
| 23 | csbeq1a 3913 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
| 24 | 6, 20, 21, 22, 23 | cbvmptf 5251 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵) | 
| 25 | 24 | fmpt 7130 | . . 3 ⊢ (∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶) | 
| 26 | 19, 25 | sylib 218 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶) | 
| 27 | fmptdF.2 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 28 | 27 | feq1i 6727 | . 2 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶) | 
| 29 | 26, 28 | sylibr 234 | 1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 [wsb 2064 ∈ wcel 2108 Ⅎwnfc 2890 ∀wral 3061 [wsbc 3788 ⦋csb 3899 ↦ cmpt 5225 ⟶wf 6557 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 df-fn 6564 df-f 6565 | 
| This theorem is referenced by: fmptcof2 32667 esumcl 34031 esumid 34045 esumgsum 34046 esumval 34047 esumel 34048 esumsplit 34054 esumaddf 34062 esumss 34073 esumpfinvalf 34077 | 
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