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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 7686 | . 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 1537 ∈ wcel 2106 ↦ cmpt 5231 ◡ccnv 5688 –1-1-onto→wf1o 6562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 |
This theorem is referenced by: f1opw2 7688 en3d 9028 f1opwfi 9394 mapfien 9446 djulf1o 9950 djurf1o 9951 fin23lem22 10365 negf1o 11691 incexclem 15869 dvdsflip 16351 hashgcdlem 16822 grplmulf1o 19044 grpraddf1o 19045 conjghm 19280 gapm 19337 sylow2a 19652 lsmhash 19738 psrbagconf1o 21967 psdmul 22188 hmeoimaf1o 23794 itg1mulc 25754 resinf1o 26593 eff1olem 26605 sqff1o 27240 dvdsppwf1o 27244 dvdsflf1o 27245 fcobij 32740 mgcf1o 32978 subfacp1lem3 35167 subfacp1lem5 35169 metakunt15 42201 metakunt16 42202 f1o2d2 42253 frlmsnic 42527 isubgr3stgrlem8 47876 |
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