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Mirrors > Home > MPE Home > Th. List > f1oexbi | Structured version Visualization version GIF version |
Description: There is a one-to-one onto function from a set to a second set iff there is a one-to-one onto function from the second set to the first set. (Contributed by Alexander van der Vekens, 30-Sep-2018.) |
Ref | Expression |
---|---|
f1oexbi | ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 ↔ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3401 | . . . . 5 ⊢ 𝑓 ∈ V | |
2 | 1 | cnvex 7392 | . . . 4 ⊢ ◡𝑓 ∈ V |
3 | f1ocnv 6403 | . . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ◡𝑓:𝐵–1-1-onto→𝐴) | |
4 | f1oeq1 6380 | . . . . 5 ⊢ (𝑔 = ◡𝑓 → (𝑔:𝐵–1-1-onto→𝐴 ↔ ◡𝑓:𝐵–1-1-onto→𝐴)) | |
5 | 4 | spcegv 3496 | . . . 4 ⊢ (◡𝑓 ∈ V → (◡𝑓:𝐵–1-1-onto→𝐴 → ∃𝑔 𝑔:𝐵–1-1-onto→𝐴)) |
6 | 2, 3, 5 | mpsyl 68 | . . 3 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ∃𝑔 𝑔:𝐵–1-1-onto→𝐴) |
7 | 6 | exlimiv 1973 | . 2 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → ∃𝑔 𝑔:𝐵–1-1-onto→𝐴) |
8 | vex 3401 | . . . . 5 ⊢ 𝑔 ∈ V | |
9 | 8 | cnvex 7392 | . . . 4 ⊢ ◡𝑔 ∈ V |
10 | f1ocnv 6403 | . . . 4 ⊢ (𝑔:𝐵–1-1-onto→𝐴 → ◡𝑔:𝐴–1-1-onto→𝐵) | |
11 | f1oeq1 6380 | . . . . 5 ⊢ (𝑓 = ◡𝑔 → (𝑓:𝐴–1-1-onto→𝐵 ↔ ◡𝑔:𝐴–1-1-onto→𝐵)) | |
12 | 11 | spcegv 3496 | . . . 4 ⊢ (◡𝑔 ∈ V → (◡𝑔:𝐴–1-1-onto→𝐵 → ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)) |
13 | 9, 10, 12 | mpsyl 68 | . . 3 ⊢ (𝑔:𝐵–1-1-onto→𝐴 → ∃𝑓 𝑓:𝐴–1-1-onto→𝐵) |
14 | 13 | exlimiv 1973 | . 2 ⊢ (∃𝑔 𝑔:𝐵–1-1-onto→𝐴 → ∃𝑓 𝑓:𝐴–1-1-onto→𝐵) |
15 | 7, 14 | impbii 201 | 1 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 ↔ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∃wex 1823 ∈ wcel 2107 Vcvv 3398 ◡ccnv 5354 –1-1-onto→wf1o 6134 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 |
This theorem is referenced by: rusgrnumwlkg 27358 f1ocnt 30123 |
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