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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 3449 | . . . 4 ⊢ 𝑚 ∈ V | |
2 | foeq1 6752 | . . . 4 ⊢ (𝑓 = 𝑚 → (𝑓:𝐴–onto→𝐵 ↔ 𝑚:𝐴–onto→𝐵)) | |
3 | 1, 2 | elab 3630 | . . 3 ⊢ (𝑚 ∈ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} ↔ 𝑚:𝐴–onto→𝐵) |
4 | fof 6756 | . . . 4 ⊢ (𝑚:𝐴–onto→𝐵 → 𝑚:𝐴⟶𝐵) | |
5 | forn 6759 | . . . . . 6 ⊢ (𝑚:𝐴–onto→𝐵 → ran 𝑚 = 𝐵) | |
6 | 1 | rnex 7849 | . . . . . 6 ⊢ ran 𝑚 ∈ V |
7 | 5, 6 | eqeltrrdi 2847 | . . . . 5 ⊢ (𝑚:𝐴–onto→𝐵 → 𝐵 ∈ V) |
8 | dmfex 7844 | . . . . . 6 ⊢ ((𝑚 ∈ V ∧ 𝑚:𝐴⟶𝐵) → 𝐴 ∈ V) | |
9 | 1, 4, 8 | sylancr 587 | . . . . 5 ⊢ (𝑚:𝐴–onto→𝐵 → 𝐴 ∈ V) |
10 | 7, 9 | elmapd 8779 | . . . 4 ⊢ (𝑚:𝐴–onto→𝐵 → (𝑚 ∈ (𝐵 ↑m 𝐴) ↔ 𝑚:𝐴⟶𝐵)) |
11 | 4, 10 | mpbird 256 | . . 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 3948 | 1 ⊢ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} ⊆ (𝐵 ↑m 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 {cab 2713 Vcvv 3445 ⊆ wss 3910 ran crn 5634 ⟶wf 6492 –onto→wfo 6494 (class class class)co 7357 ↑m cmap 8765 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-fo 6502 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 df-map 8767 |
This theorem is referenced by: fosetex 8796 |
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