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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 2910 | . . . . . 6 ⊢ (𝐶 ∈ 𝐵 → (𝑦 = 𝐶 → 𝑦 ∈ 𝐵)) | |
5 | 2, 4 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = 𝐶 → 𝑦 ∈ 𝐵)) |
6 | 5 | impr 457 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)) → 𝑦 ∈ 𝐵) |
7 | f1o2d.4 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶)) | |
8 | 7 | biimpar 480 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑦 = 𝐶) → 𝑥 = 𝐷) |
9 | 8 | exp42 438 | . . . . . 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 421 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)) → (𝑦 ∈ 𝐵 → 𝑥 = 𝐷)) |
12 | 6, 11 | jcai 519 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)) → (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)) |
13 | eleq1a 2910 | . . . . . 6 ⊢ (𝐷 ∈ 𝐴 → (𝑥 = 𝐷 → 𝑥 ∈ 𝐴)) | |
14 | 3, 13 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑥 = 𝐷 → 𝑥 ∈ 𝐴)) |
15 | 14 | impr 457 | . . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)) → 𝑥 ∈ 𝐴) |
16 | 7 | biimpa 479 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑥 = 𝐷) → 𝑦 = 𝐶) |
17 | 16 | exp42 438 | . . . . . . 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 421 | . . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)) → (𝑥 ∈ 𝐴 → 𝑦 = 𝐶)) |
21 | 15, 20 | jcai 519 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)) → (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)) |
22 | 12, 21 | impbida 799 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷))) |
23 | 1, 2, 3, 22 | f1ocnvd 7398 | 1 ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ∧ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ↦ cmpt 5148 ◡ccnv 5556 –1-1-onto→wf1o 6356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 |
This theorem is referenced by: f1o2d 7401 negiso 11623 iccf1o 12885 bitsf1ocnv 15795 grpinvcnv 18169 grplactcnv 18204 issrngd 19634 opncldf1 21694 txhmeo 22413 ptuncnv 22417 icopnfcnv 23548 iccpnfcnv 23550 xrge0iifcnv 31178 rfovcnvf1od 40357 |
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