Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > f1ocnvd | Structured version Visualization version GIF version |
Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
f1od.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) |
f1od.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑊) |
f1od.3 | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝑋) |
f1od.4 | ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷))) |
Ref | Expression |
---|---|
f1ocnvd | ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ∧ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1od.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑊) | |
2 | 1 | ralrimiva 2949 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝑊) |
3 | f1od.1 | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
4 | 3 | fnmpt 5933 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝑊 → 𝐹 Fn 𝐴) |
5 | 2, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
6 | f1od.3 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝑋) | |
7 | 6 | ralrimiva 2949 | . . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 𝐷 ∈ 𝑋) |
8 | eqid 2610 | . . . . . 6 ⊢ (𝑦 ∈ 𝐵 ↦ 𝐷) = (𝑦 ∈ 𝐵 ↦ 𝐷) | |
9 | 8 | fnmpt 5933 | . . . . 5 ⊢ (∀𝑦 ∈ 𝐵 𝐷 ∈ 𝑋 → (𝑦 ∈ 𝐵 ↦ 𝐷) Fn 𝐵) |
10 | 7, 9 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ 𝐷) Fn 𝐵) |
11 | f1od.4 | . . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷))) | |
12 | 11 | opabbidv 4648 | . . . . . 6 ⊢ (𝜑 → {〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)}) |
13 | df-mpt 4645 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} | |
14 | 3, 13 | eqtri 2632 | . . . . . . . 8 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} |
15 | 14 | cnveqi 5219 | . . . . . . 7 ⊢ ◡𝐹 = ◡{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} |
16 | cnvopab 5452 | . . . . . . 7 ⊢ ◡{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} = {〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} | |
17 | 15, 16 | eqtri 2632 | . . . . . 6 ⊢ ◡𝐹 = {〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} |
18 | df-mpt 4645 | . . . . . 6 ⊢ (𝑦 ∈ 𝐵 ↦ 𝐷) = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)} | |
19 | 12, 17, 18 | 3eqtr4g 2669 | . . . . 5 ⊢ (𝜑 → ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷)) |
20 | 19 | fneq1d 5895 | . . . 4 ⊢ (𝜑 → (◡𝐹 Fn 𝐵 ↔ (𝑦 ∈ 𝐵 ↦ 𝐷) Fn 𝐵)) |
21 | 10, 20 | mpbird 246 | . . 3 ⊢ (𝜑 → ◡𝐹 Fn 𝐵) |
22 | dff1o4 6058 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵)) | |
23 | 5, 21, 22 | sylanbrc 695 | . 2 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) |
24 | 23, 19 | jca 553 | 1 ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ∧ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {copab 4642 ↦ cmpt 4643 ◡ccnv 5037 Fn wfn 5799 –1-1-onto→wf1o 5803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 |
This theorem is referenced by: f1od 6783 f1ocnv2d 6784 pw2f1ocnv 36622 |
Copyright terms: Public domain | W3C validator |