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Mirrors > Home > MPE Home > Th. List > permsetexOLD | Structured version Visualization version GIF version |
Description: Obsolete version of f1osetex 8529 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 8503 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → {𝑓 ∣ 𝑓:𝐴⟶𝐴} ∈ V) | |
2 | 1 | anidms 570 | . 2 ⊢ (𝐴 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴⟶𝐴} ∈ V) |
3 | f1of 6650 | . . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐴 → 𝑓:𝐴⟶𝐴) | |
4 | 3 | ss2abi 3970 | . . 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 5206 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 {cab 2712 Vcvv 3401 ⊆ wss 3857 ⟶wf 6365 –1-1-onto→wf1o 6368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-xp 5546 df-rel 5547 df-cnv 5548 df-dm 5550 df-rn 5551 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-f1o 6376 |
This theorem is referenced by: symgbasexOLD 18736 |
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