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| Mirrors > Home > MPE Home > Th. List > f1ocnv2d | 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) | 
| f1o2d.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) | 
| f1o2d.3 | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝐴) | 
| f1o2d.4 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶)) | 
| Ref | Expression | 
|---|---|
| f1ocnv2d | ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ∧ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | f1od.1 | . 2 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
| 2 | f1o2d.2 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) | |
| 3 | f1o2d.3 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝐴) | |
| 4 | eleq1a 2835 | . . . . . 6 ⊢ (𝐶 ∈ 𝐵 → (𝑦 = 𝐶 → 𝑦 ∈ 𝐵)) | |
| 5 | 2, 4 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = 𝐶 → 𝑦 ∈ 𝐵)) | 
| 6 | 5 | impr 454 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)) → 𝑦 ∈ 𝐵) | 
| 7 | f1o2d.4 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶)) | |
| 8 | 7 | biimpar 477 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑦 = 𝐶) → 𝑥 = 𝐷) | 
| 9 | 8 | exp42 435 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → (𝑦 = 𝐶 → 𝑥 = 𝐷)))) | 
| 10 | 9 | com34 91 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑦 = 𝐶 → (𝑦 ∈ 𝐵 → 𝑥 = 𝐷)))) | 
| 11 | 10 | imp32 418 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)) → (𝑦 ∈ 𝐵 → 𝑥 = 𝐷)) | 
| 12 | 6, 11 | jcai 516 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)) → (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)) | 
| 13 | eleq1a 2835 | . . . . . 6 ⊢ (𝐷 ∈ 𝐴 → (𝑥 = 𝐷 → 𝑥 ∈ 𝐴)) | |
| 14 | 3, 13 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑥 = 𝐷 → 𝑥 ∈ 𝐴)) | 
| 15 | 14 | impr 454 | . . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)) → 𝑥 ∈ 𝐴) | 
| 16 | 7 | biimpa 476 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑥 = 𝐷) → 𝑦 = 𝐶) | 
| 17 | 16 | exp42 435 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → (𝑥 = 𝐷 → 𝑦 = 𝐶)))) | 
| 18 | 17 | com23 86 | . . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → (𝑥 ∈ 𝐴 → (𝑥 = 𝐷 → 𝑦 = 𝐶)))) | 
| 19 | 18 | com34 91 | . . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → (𝑥 = 𝐷 → (𝑥 ∈ 𝐴 → 𝑦 = 𝐶)))) | 
| 20 | 19 | imp32 418 | . . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)) → (𝑥 ∈ 𝐴 → 𝑦 = 𝐶)) | 
| 21 | 15, 20 | jcai 516 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)) → (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)) | 
| 22 | 12, 21 | impbida 800 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷))) | 
| 23 | 1, 2, 3, 22 | f1ocnvd 7685 | 1 ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ∧ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ↦ cmpt 5224 ◡ccnv 5683 –1-1-onto→wf1o 6559 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 | 
| This theorem is referenced by: f1o2d 7688 negiso 12249 iccf1o 13537 bitsf1ocnv 16482 grpinvcnv 19025 grplactcnv 19062 issrngd 20857 opncldf1 23093 txhmeo 23812 ptuncnv 23816 icopnfcnv 24974 iccpnfcnv 24976 gsumwrd2dccatlem 33070 xrge0iifcnv 33933 rfovcnvf1od 44022 | 
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