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Mirrors > Home > MPE Home > Th. List > fosetex | Structured version Visualization version GIF version |
Description: The set of surjections between two classes exists (without any precondition). (Contributed by AV, 8-Aug-2024.) |
Ref | Expression |
---|---|
fosetex | ⊢ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7288 | . 2 ⊢ (𝐵 ↑m 𝐴) ∈ V | |
2 | mapfoss 8598 | . 2 ⊢ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} ⊆ (𝐵 ↑m 𝐴) | |
3 | 1, 2 | ssexi 5241 | 1 ⊢ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2108 {cab 2715 Vcvv 3422 –onto→wfo 6416 (class class class)co 7255 ↑m cmap 8573 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-sbc 3712 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-fo 6424 df-fv 6426 df-ov 7258 df-oprab 7259 df-mpo 7260 df-map 8575 |
This theorem is referenced by: f1osetex 8605 |
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