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 6873 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 2075 | . . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) → [𝑦 / 𝑥]𝐵 ∈ 𝐶) |
3 | sban 2082 | . . . . . 6 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝑥 ∈ 𝐴)) | |
4 | fmptdF.p | . . . . . . . 8 ⊢ Ⅎ𝑥𝜑 | |
5 | 4 | sbf 2266 | . . . . . . 7 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑) |
6 | fmptdF.a | . . . . . . . 8 ⊢ Ⅎ𝑥𝐴 | |
7 | 6 | clelsb3fw 2981 | . . . . . . 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 277 | . . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑦 ∈ 𝐴)) |
10 | sbsbc 3776 | . . . . . 6 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ [𝑦 / 𝑥]𝐵 ∈ 𝐶) | |
11 | sbcel12 4360 | . . . . . . 7 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ ⦋𝑦 / 𝑥⦌𝐶) | |
12 | vex 3498 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
13 | fmptdF.c | . . . . . . . . 9 ⊢ Ⅎ𝑥𝐶 | |
14 | 12, 13 | csbgfi 3903 | . . . . . . . 8 ⊢ ⦋𝑦 / 𝑥⦌𝐶 = 𝐶 |
15 | 14 | eleq2i 2904 | . . . . . . 7 ⊢ (⦋𝑦 / 𝑥⦌𝐵 ∈ ⦋𝑦 / 𝑥⦌𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶) |
16 | 11, 15 | bitri 277 | . . . . . 6 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶) |
17 | 10, 16 | bitri 277 | . . . . 5 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶) |
18 | 2, 9, 17 | 3imtr3i 293 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶) |
19 | 18 | ralrimiva 3182 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶) |
20 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
21 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑦𝐵 | |
22 | nfcsb1v 3907 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
23 | csbeq1a 3897 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
24 | 6, 20, 21, 22, 23 | cbvmptf 5158 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵) |
25 | 24 | fmpt 6869 | . . 3 ⊢ (∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶) |
26 | 19, 25 | sylib 220 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶) |
27 | fmptdF.2 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
28 | 27 | feq1i 6500 | . 2 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶) |
29 | 26, 28 | sylibr 236 | 1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 Ⅎwnf 1780 [wsb 2065 ∈ wcel 2110 Ⅎwnfc 2961 ∀wral 3138 [wsbc 3772 ⦋csb 3883 ↦ cmpt 5139 ⟶wf 6346 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-fv 6358 |
This theorem is referenced by: fmptcof2 30396 esumcl 31284 esumid 31298 esumgsum 31299 esumval 31300 esumel 31301 esumsplit 31307 esumaddf 31315 esumss 31326 esumpfinvalf 31330 |
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