Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > feqmptdf | Structured version Visualization version GIF version |
Description: Deduction form of dffn5f 6252. (Contributed by Mario Carneiro, 8-Jan-2015.) (Revised by Thierry Arnoux, 10-May-2017.) |
Ref | Expression |
---|---|
feqmptdf.1 | ⊢ Ⅎ𝑥𝐴 |
feqmptdf.2 | ⊢ Ⅎ𝑥𝐹 |
feqmptdf.3 | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
Ref | Expression |
---|---|
feqmptdf | ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | feqmptdf.3 | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
2 | ffn 6045 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
3 | fnrel 5989 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
4 | feqmptdf.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 | |
5 | nfcv 2764 | . . . . . 6 ⊢ Ⅎ𝑦𝐹 | |
6 | 4, 5 | dfrel4 5585 | . . . . 5 ⊢ (Rel 𝐹 ↔ 𝐹 = {〈𝑥, 𝑦〉 ∣ 𝑥𝐹𝑦}) |
7 | 3, 6 | sylib 208 | . . . 4 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {〈𝑥, 𝑦〉 ∣ 𝑥𝐹𝑦}) |
8 | feqmptdf.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
9 | 4, 8 | nffn 5987 | . . . . 5 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴 |
10 | nfv 1843 | . . . . 5 ⊢ Ⅎ𝑦 𝐹 Fn 𝐴 | |
11 | fnbr 5993 | . . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥𝐹𝑦) → 𝑥 ∈ 𝐴) | |
12 | 11 | ex 450 | . . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 → 𝑥 ∈ 𝐴)) |
13 | 12 | pm4.71rd 667 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))) |
14 | eqcom 2629 | . . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦) | |
15 | fnbrfvb 6236 | . . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦)) | |
16 | 14, 15 | syl5bb 272 | . . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦)) |
17 | 16 | pm5.32da 673 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))) |
18 | 13, 17 | bitr4d 271 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)))) |
19 | 9, 10, 18 | opabbid 4715 | . . . 4 ⊢ (𝐹 Fn 𝐴 → {〈𝑥, 𝑦〉 ∣ 𝑥𝐹𝑦} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))}) |
20 | 7, 19 | eqtrd 2656 | . . 3 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))}) |
21 | df-mpt 4730 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))} | |
22 | 20, 21 | syl6eqr 2674 | . 2 ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
23 | 1, 2, 22 | 3syl 18 | 1 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Ⅎwnfc 2751 class class class wbr 4653 {copab 4712 ↦ cmpt 4729 Rel wrel 5119 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 |
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-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-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 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-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 |
This theorem is referenced by: esumf1o 30112 feqresmptf 39433 liminfvaluz3 40028 liminfvaluz4 40031 volioofmpt 40211 volicofmpt 40214 |
Copyright terms: Public domain | W3C validator |