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Mirrors > Home > NFE Home > Th. List > f1o2d | 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 | . 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 2422 | . . . . . 6 ⊢ (C ∈ B → (y = C → y ∈ B)) | |
5 | 2, 4 | syl 15 | . . . . 5 ⊢ ((φ ∧ x ∈ A) → (y = C → y ∈ B)) |
6 | 5 | impr 602 | . . . 4 ⊢ ((φ ∧ (x ∈ A ∧ y = C)) → y ∈ B) |
7 | f1o2d.4 | . . . . . . . 8 ⊢ ((φ ∧ (x ∈ A ∧ y ∈ B)) → (x = D ↔ y = C)) | |
8 | 7 | biimpar 471 | . . . . . . 7 ⊢ (((φ ∧ (x ∈ A ∧ y ∈ B)) ∧ y = C) → x = D) |
9 | 8 | exp42 594 | . . . . . 6 ⊢ (φ → (x ∈ A → (y ∈ B → (y = C → x = D)))) |
10 | 9 | com34 77 | . . . . 5 ⊢ (φ → (x ∈ A → (y = C → (y ∈ B → x = D)))) |
11 | 10 | imp32 422 | . . . 4 ⊢ ((φ ∧ (x ∈ A ∧ y = C)) → (y ∈ B → x = D)) |
12 | 6, 11 | jcai 522 | . . 3 ⊢ ((φ ∧ (x ∈ A ∧ y = C)) → (y ∈ B ∧ x = D)) |
13 | eleq1a 2422 | . . . . . 6 ⊢ (D ∈ A → (x = D → x ∈ A)) | |
14 | 3, 13 | syl 15 | . . . . 5 ⊢ ((φ ∧ y ∈ B) → (x = D → x ∈ A)) |
15 | 14 | impr 602 | . . . 4 ⊢ ((φ ∧ (y ∈ B ∧ x = D)) → x ∈ A) |
16 | 7 | biimpa 470 | . . . . . . . 8 ⊢ (((φ ∧ (x ∈ A ∧ y ∈ B)) ∧ x = D) → y = C) |
17 | 16 | exp42 594 | . . . . . . 7 ⊢ (φ → (x ∈ A → (y ∈ B → (x = D → y = C)))) |
18 | 17 | com23 72 | . . . . . 6 ⊢ (φ → (y ∈ B → (x ∈ A → (x = D → y = C)))) |
19 | 18 | com34 77 | . . . . 5 ⊢ (φ → (y ∈ B → (x = D → (x ∈ A → y = C)))) |
20 | 19 | imp32 422 | . . . 4 ⊢ ((φ ∧ (y ∈ B ∧ x = D)) → (x ∈ A → y = C)) |
21 | 15, 20 | jcai 522 | . . 3 ⊢ ((φ ∧ (y ∈ B ∧ x = D)) → (x ∈ A ∧ y = C)) |
22 | 12, 21 | impbida 805 | . 2 ⊢ (φ → ((x ∈ A ∧ y = C) ↔ (y ∈ B ∧ x = D))) |
23 | 1, 2, 3, 22 | f1od 5726 | 1 ⊢ (φ → F:A–1-1-onto→B) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 = wceq 1642 ∈ wcel 1710 –1-1-onto→wf1o 4780 ↦ cmpt 5651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-co 4726 df-ima 4727 df-id 4767 df-cnv 4785 df-rn 4786 df-dm 4787 df-fun 4789 df-fn 4790 df-f 4791 df-f1 4792 df-fo 4793 df-f1o 4794 df-mpt 5652 |
This theorem is referenced by: (None) |
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