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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 3486 | . . . 4 ⊢ 𝑚 ∈ V | |
2 | foeq1 6829 | . . . 4 ⊢ (𝑓 = 𝑚 → (𝑓:𝐴–onto→𝐵 ↔ 𝑚:𝐴–onto→𝐵)) | |
3 | 1, 2 | elab 3689 | . . 3 ⊢ (𝑚 ∈ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} ↔ 𝑚:𝐴–onto→𝐵) |
4 | fof 6833 | . . . 4 ⊢ (𝑚:𝐴–onto→𝐵 → 𝑚:𝐴⟶𝐵) | |
5 | forn 6836 | . . . . . 6 ⊢ (𝑚:𝐴–onto→𝐵 → ran 𝑚 = 𝐵) | |
6 | 1 | rnex 7946 | . . . . . 6 ⊢ ran 𝑚 ∈ V |
7 | 5, 6 | eqeltrrdi 2847 | . . . . 5 ⊢ (𝑚:𝐴–onto→𝐵 → 𝐵 ∈ V) |
8 | dmfex 7941 | . . . . . 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 8894 | . . . 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 217 | . 2 ⊢ (𝑚 ∈ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} → 𝑚 ∈ (𝐵 ↑m 𝐴)) |
13 | 12 | ssriv 4006 | 1 ⊢ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} ⊆ (𝐵 ↑m 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2103 {cab 2711 Vcvv 3482 ⊆ wss 3970 ran crn 5700 ⟶wf 6568 –onto→wfo 6570 (class class class)co 7445 ↑m cmap 8880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-sbc 3799 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5170 df-opab 5232 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-fo 6578 df-fv 6580 df-ov 7448 df-oprab 7449 df-mpo 7450 df-map 8882 |
This theorem is referenced by: fosetex 8912 |
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