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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 3403 | . . . . 5 ⊢ 𝑓 ∈ V | |
2 | 1 | cnvex 7659 | . . . 4 ⊢ ◡𝑓 ∈ V |
3 | f1ocnv 6633 | . . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ◡𝑓:𝐵–1-1-onto→𝐴) | |
4 | f1oeq1 6609 | . . . . 5 ⊢ (𝑔 = ◡𝑓 → (𝑔:𝐵–1-1-onto→𝐴 ↔ ◡𝑓:𝐵–1-1-onto→𝐴)) | |
5 | 4 | spcegv 3502 | . . . 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 1937 | . 2 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → ∃𝑔 𝑔:𝐵–1-1-onto→𝐴) |
8 | vex 3403 | . . . . 5 ⊢ 𝑔 ∈ V | |
9 | 8 | cnvex 7659 | . . . 4 ⊢ ◡𝑔 ∈ V |
10 | f1ocnv 6633 | . . . 4 ⊢ (𝑔:𝐵–1-1-onto→𝐴 → ◡𝑔:𝐴–1-1-onto→𝐵) | |
11 | f1oeq1 6609 | . . . . 5 ⊢ (𝑓 = ◡𝑔 → (𝑓:𝐴–1-1-onto→𝐵 ↔ ◡𝑔:𝐴–1-1-onto→𝐵)) | |
12 | 11 | spcegv 3502 | . . . 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 1937 | . 2 ⊢ (∃𝑔 𝑔:𝐵–1-1-onto→𝐴 → ∃𝑓 𝑓:𝐴–1-1-onto→𝐵) |
15 | 7, 14 | impbii 212 | 1 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 ↔ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∃wex 1786 ∈ wcel 2114 Vcvv 3399 ◡ccnv 5525 –1-1-onto→wf1o 6339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-12 2179 ax-ext 2711 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3401 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-br 5032 df-opab 5094 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 |
This theorem is referenced by: rusgrnumwlkg 27918 f1ocnt 30701 |
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