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Mirrors > Home > MPE Home > Th. List > f1osetex | Structured version Visualization version GIF version |
Description: The set of bijections between two classes exists. (Contributed by AV, 30-Mar-2024.) (Revised by AV, 8-Aug-2024.) (Proof shortened by SN, 22-Aug-2024.) |
Ref | Expression |
---|---|
f1osetex | ⊢ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐵} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fosetex 8453 | . 2 ⊢ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} ∈ V | |
2 | f1ofo 6614 | . . 3 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵) | |
3 | 2 | ss2abi 3973 | . 2 ⊢ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐵} ⊆ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} |
4 | 1, 3 | ssexi 5196 | 1 ⊢ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐵} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 {cab 2735 Vcvv 3409 –onto→wfo 6338 –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 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ral 3075 df-rex 3076 df-rab 3079 df-v 3411 df-sbc 3699 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4802 df-br 5037 df-opab 5099 df-id 5434 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-ov 7159 df-oprab 7160 df-mpo 7161 df-map 8424 |
This theorem is referenced by: hashfacen 13875 symgplusg 18592 symgvalstruct 18606 |
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