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Mirrors > Home > MPE Home > Th. List > permsetexOLD | Structured version Visualization version GIF version |
Description: Obsolete version of f1osetex 8871 as of 8-Aug-2024. (Contributed by AV, 30-Mar-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
permsetexOLD | ⊢ (𝐴 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapex 8844 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → {𝑓 ∣ 𝑓:𝐴⟶𝐴} ∈ V) | |
2 | 1 | anidms 565 | . 2 ⊢ (𝐴 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴⟶𝐴} ∈ V) |
3 | f1of 6832 | . . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐴 → 𝑓:𝐴⟶𝐴) | |
4 | 3 | ss2abi 4056 | . . 3 ⊢ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐴} |
5 | 4 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐴}) |
6 | 2, 5 | ssexd 5320 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 {cab 2702 Vcvv 3463 ⊆ wss 3941 ⟶wf 6539 –1-1-onto→wf1o 6542 |
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-ext 2696 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 |
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-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5145 df-opab 5207 df-xp 5679 df-rel 5680 df-cnv 5681 df-dm 5683 df-rn 5684 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-f1o 6550 |
This theorem is referenced by: symgbasexOLD 19325 |
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