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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 3492 | . . . 4 ⊢ 𝑚 ∈ V | |
2 | foeq1 6830 | . . . 4 ⊢ (𝑓 = 𝑚 → (𝑓:𝐴–onto→𝐵 ↔ 𝑚:𝐴–onto→𝐵)) | |
3 | 1, 2 | elab 3694 | . . 3 ⊢ (𝑚 ∈ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} ↔ 𝑚:𝐴–onto→𝐵) |
4 | fof 6834 | . . . 4 ⊢ (𝑚:𝐴–onto→𝐵 → 𝑚:𝐴⟶𝐵) | |
5 | forn 6837 | . . . . . 6 ⊢ (𝑚:𝐴–onto→𝐵 → ran 𝑚 = 𝐵) | |
6 | 1 | rnex 7950 | . . . . . 6 ⊢ ran 𝑚 ∈ V |
7 | 5, 6 | eqeltrrdi 2853 | . . . . 5 ⊢ (𝑚:𝐴–onto→𝐵 → 𝐵 ∈ V) |
8 | dmfex 7945 | . . . . . 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 8898 | . . . 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 4012 | 1 ⊢ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} ⊆ (𝐵 ↑m 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2108 {cab 2717 Vcvv 3488 ⊆ wss 3976 ran crn 5701 ⟶wf 6569 –onto→wfo 6571 (class class class)co 7448 ↑m cmap 8884 |
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 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 |
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 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-fo 6579 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-map 8886 |
This theorem is referenced by: fosetex 8916 |
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