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| Mirrors > Home > MPE Home > Th. List > feqmptdf | Structured version Visualization version GIF version | ||
| Description: Deduction form of dffn5f 6980. (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 6736 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
| 3 | fnrel 6670 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
| 4 | feqmptdf.2 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 | |
| 5 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑦𝐹 | |
| 6 | 4, 5 | dfrel4 6211 | . . . . 5 ⊢ (Rel 𝐹 ↔ 𝐹 = {〈𝑥, 𝑦〉 ∣ 𝑥𝐹𝑦}) |
| 7 | 3, 6 | sylib 218 | . . . 4 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {〈𝑥, 𝑦〉 ∣ 𝑥𝐹𝑦}) |
| 8 | feqmptdf.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
| 9 | 4, 8 | nffn 6667 | . . . . 5 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴 |
| 10 | nfv 1914 | . . . . 5 ⊢ Ⅎ𝑦 𝐹 Fn 𝐴 | |
| 11 | fnbr 6676 | . . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥𝐹𝑦) → 𝑥 ∈ 𝐴) | |
| 12 | 11 | ex 412 | . . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 → 𝑥 ∈ 𝐴)) |
| 13 | 12 | pm4.71rd 562 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))) |
| 14 | eqcom 2744 | . . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦) | |
| 15 | fnbrfvb 6959 | . . . . . . . 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 579 | . . . . . 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 5208 | . . . 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 5226 | . . 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 395 = wceq 1540 ∈ wcel 2108 Ⅎwnfc 2890 class class class wbr 5143 {copab 5205 ↦ cmpt 5225 Rel wrel 5690 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 |
| 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-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 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-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 |
| This theorem is referenced by: esumf1o 34051 feqresmptf 45236 liminfvaluz3 45811 liminfvaluz4 45814 volioofmpt 46009 volicofmpt 46012 |
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