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Mirrors > Home > MPE Home > Th. List > feqmptdf | Structured version Visualization version GIF version |
Description: Deduction form of dffn5f 6908. (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 6663 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
3 | fnrel 6599 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
4 | feqmptdf.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 | |
5 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑦𝐹 | |
6 | 4, 5 | dfrel4 6139 | . . . . 5 ⊢ (Rel 𝐹 ↔ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦}) |
7 | 3, 6 | sylib 217 | . . . 4 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦}) |
8 | feqmptdf.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
9 | 4, 8 | nffn 6596 | . . . . 5 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴 |
10 | nfv 1917 | . . . . 5 ⊢ Ⅎ𝑦 𝐹 Fn 𝐴 | |
11 | fnbr 6605 | . . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥𝐹𝑦) → 𝑥 ∈ 𝐴) | |
12 | 11 | ex 413 | . . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 → 𝑥 ∈ 𝐴)) |
13 | 12 | pm4.71rd 563 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))) |
14 | eqcom 2744 | . . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦) | |
15 | fnbrfvb 6890 | . . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦)) | |
16 | 14, 15 | bitrid 282 | . . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦)) |
17 | 16 | pm5.32da 579 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))) |
18 | 13, 17 | bitr4d 281 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)))) |
19 | 9, 10, 18 | opabbid 5168 | . . . 4 ⊢ (𝐹 Fn 𝐴 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))}) |
20 | 7, 19 | eqtrd 2777 | . . 3 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))}) |
21 | df-mpt 5187 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))} | |
22 | 20, 21 | eqtr4di 2795 | . 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 396 = wceq 1541 ∈ wcel 2106 Ⅎwnfc 2885 class class class wbr 5103 {copab 5165 ↦ cmpt 5186 Rel wrel 5635 Fn wfn 6486 ⟶wf 6487 ‘cfv 6491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5528 df-xp 5636 df-rel 5637 df-cnv 5638 df-co 5639 df-dm 5640 df-iota 6443 df-fun 6493 df-fn 6494 df-f 6495 df-fv 6499 |
This theorem is referenced by: esumf1o 32422 feqresmptf 43211 liminfvaluz3 43790 liminfvaluz4 43793 volioofmpt 43988 volicofmpt 43991 |
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