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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 3440 | . . . . 5 ⊢ 𝑓 ∈ V | |
2 | 1 | cnvex 7491 | . . . 4 ⊢ ◡𝑓 ∈ V |
3 | f1ocnv 6500 | . . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ◡𝑓:𝐵–1-1-onto→𝐴) | |
4 | f1oeq1 6477 | . . . . 5 ⊢ (𝑔 = ◡𝑓 → (𝑔:𝐵–1-1-onto→𝐴 ↔ ◡𝑓:𝐵–1-1-onto→𝐴)) | |
5 | 4 | spcegv 3540 | . . . 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 1908 | . 2 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → ∃𝑔 𝑔:𝐵–1-1-onto→𝐴) |
8 | vex 3440 | . . . . 5 ⊢ 𝑔 ∈ V | |
9 | 8 | cnvex 7491 | . . . 4 ⊢ ◡𝑔 ∈ V |
10 | f1ocnv 6500 | . . . 4 ⊢ (𝑔:𝐵–1-1-onto→𝐴 → ◡𝑔:𝐴–1-1-onto→𝐵) | |
11 | f1oeq1 6477 | . . . . 5 ⊢ (𝑓 = ◡𝑔 → (𝑓:𝐴–1-1-onto→𝐵 ↔ ◡𝑔:𝐴–1-1-onto→𝐵)) | |
12 | 11 | spcegv 3540 | . . . 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 1908 | . 2 ⊢ (∃𝑔 𝑔:𝐵–1-1-onto→𝐴 → ∃𝑓 𝑓:𝐴–1-1-onto→𝐵) |
15 | 7, 14 | impbii 210 | 1 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 ↔ ∃𝑔 𝑔:𝐵–1-1-onto→𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∃wex 1761 ∈ wcel 2081 Vcvv 3437 ◡ccnv 5447 –1-1-onto→wf1o 6229 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-op 4483 df-uni 4750 df-br 4967 df-opab 5029 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 |
This theorem is referenced by: rusgrnumwlkg 27448 f1ocnt 30214 |
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