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Mirrors > Home > MPE Home > Th. List > mapfoss | Structured version Visualization version GIF version |
Description: The value of the set exponentiation (𝐵 ↑m 𝐴) is a superset of the set of all functions from 𝐴 onto 𝐵. (Contributed by AV, 7-Aug-2024.) |
Ref | Expression |
---|---|
mapfoss | ⊢ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} ⊆ (𝐵 ↑m 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3472 | . . . 4 ⊢ 𝑚 ∈ V | |
2 | foeq1 6794 | . . . 4 ⊢ (𝑓 = 𝑚 → (𝑓:𝐴–onto→𝐵 ↔ 𝑚:𝐴–onto→𝐵)) | |
3 | 1, 2 | elab 3663 | . . 3 ⊢ (𝑚 ∈ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} ↔ 𝑚:𝐴–onto→𝐵) |
4 | fof 6798 | . . . 4 ⊢ (𝑚:𝐴–onto→𝐵 → 𝑚:𝐴⟶𝐵) | |
5 | forn 6801 | . . . . . 6 ⊢ (𝑚:𝐴–onto→𝐵 → ran 𝑚 = 𝐵) | |
6 | 1 | rnex 7899 | . . . . . 6 ⊢ ran 𝑚 ∈ V |
7 | 5, 6 | eqeltrrdi 2836 | . . . . 5 ⊢ (𝑚:𝐴–onto→𝐵 → 𝐵 ∈ V) |
8 | dmfex 7894 | . . . . . 6 ⊢ ((𝑚 ∈ V ∧ 𝑚:𝐴⟶𝐵) → 𝐴 ∈ V) | |
9 | 1, 4, 8 | sylancr 586 | . . . . 5 ⊢ (𝑚:𝐴–onto→𝐵 → 𝐴 ∈ V) |
10 | 7, 9 | elmapd 8833 | . . . 4 ⊢ (𝑚:𝐴–onto→𝐵 → (𝑚 ∈ (𝐵 ↑m 𝐴) ↔ 𝑚:𝐴⟶𝐵)) |
11 | 4, 10 | mpbird 257 | . . 3 ⊢ (𝑚:𝐴–onto→𝐵 → 𝑚 ∈ (𝐵 ↑m 𝐴)) |
12 | 3, 11 | sylbi 216 | . 2 ⊢ (𝑚 ∈ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} → 𝑚 ∈ (𝐵 ↑m 𝐴)) |
13 | 12 | ssriv 3981 | 1 ⊢ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} ⊆ (𝐵 ↑m 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 {cab 2703 Vcvv 3468 ⊆ wss 3943 ran crn 5670 ⟶wf 6532 –onto→wfo 6534 (class class class)co 7404 ↑m cmap 8819 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fo 6542 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-map 8821 |
This theorem is referenced by: fosetex 8851 |
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