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| Mirrors > Home > MPE Home > Th. List > f1o2d | Structured version Visualization version GIF version | ||
| Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014.) | 
| 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 | 
|---|---|
| f1o2d | ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | f1od.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
| 2 | f1o2d.2 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) | |
| 3 | f1o2d.3 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝐴) | |
| 4 | f1o2d.4 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶)) | |
| 5 | 1, 2, 3, 4 | f1ocnv2d 7687 | . 2 ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ∧ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷))) | 
| 6 | 5 | simpld 494 | 1 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) | 
| 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: f1opw2 7689 en3d 9030 f1opwfi 9397 mapfien 9449 djulf1o 9953 djurf1o 9954 fin23lem22 10368 negf1o 11694 incexclem 15873 dvdsflip 16355 hashgcdlem 16826 grplmulf1o 19032 grpraddf1o 19033 conjghm 19268 gapm 19325 sylow2a 19638 lsmhash 19724 psrbagconf1o 21950 psdmul 22171 hmeoimaf1o 23779 itg1mulc 25740 resinf1o 26579 eff1olem 26591 sqff1o 27226 dvdsppwf1o 27230 dvdsflf1o 27231 fcobij 32734 mgcf1o 32994 subfacp1lem3 35188 subfacp1lem5 35190 metakunt15 42221 metakunt16 42222 f1o2d2 42274 frlmsnic 42555 isubgr3stgrlem8 47945 | 
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