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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 8883 | . 2 ⊢ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} ∈ V | |
2 | f1ofo 6851 | . . 3 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵) | |
3 | 2 | ss2abi 4063 | . 2 ⊢ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐵} ⊆ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} |
4 | 1, 3 | ssexi 5326 | 1 ⊢ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐵} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 {cab 2705 Vcvv 3473 –onto→wfo 6551 –1-1-onto→wf1o 6552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-map 8853 |
This theorem is referenced by: hashfacen 14453 symgplusg 19344 symgvalstruct 19358 symgvalstructOLD 19359 |
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