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Mirrors > Home > MPE Home > Th. List > feqmptdf | Structured version Visualization version GIF version |
Description: Deduction form of dffn5f 6909. (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 6664 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
3 | fnrel 6600 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
4 | feqmptdf.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 | |
5 | nfcv 2906 | . . . . . 6 ⊢ Ⅎ𝑦𝐹 | |
6 | 4, 5 | dfrel4 6140 | . . . . 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 6597 | . . . . 5 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴 |
10 | nfv 1918 | . . . . 5 ⊢ Ⅎ𝑦 𝐹 Fn 𝐴 | |
11 | fnbr 6606 | . . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥𝐹𝑦) → 𝑥 ∈ 𝐴) | |
12 | 11 | ex 414 | . . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 → 𝑥 ∈ 𝐴)) |
13 | 12 | pm4.71rd 564 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))) |
14 | eqcom 2745 | . . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦) | |
15 | fnbrfvb 6891 | . . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦)) | |
16 | 14, 15 | bitrid 283 | . . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦)) |
17 | 16 | pm5.32da 580 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))) |
18 | 13, 17 | bitr4d 282 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)))) |
19 | 9, 10, 18 | opabbid 5169 | . . . 4 ⊢ (𝐹 Fn 𝐴 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))}) |
20 | 7, 19 | eqtrd 2778 | . . 3 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))}) |
21 | df-mpt 5188 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))} | |
22 | 20, 21 | eqtr4di 2796 | . 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 397 = wceq 1542 ∈ wcel 2107 Ⅎwnfc 2886 class class class wbr 5104 {copab 5166 ↦ cmpt 5187 Rel wrel 5636 Fn wfn 6487 ⟶wf 6488 ‘cfv 6492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-fv 6500 |
This theorem is referenced by: esumf1o 32410 feqresmptf 43180 liminfvaluz3 43759 liminfvaluz4 43762 volioofmpt 43957 volicofmpt 43960 |
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