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Mirrors > Home > MPE Home > Th. List > mapvalg | Structured version Visualization version GIF version |
Description: The value of set exponentiation. (𝐴 ↑m 𝐵) is the set of all functions that map from 𝐵 to 𝐴. Definition 10.24 of [Kunen] p. 24. (Contributed by NM, 8-Dec-2003.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
mapvalg | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapex 8828 | . . 3 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝐴 ∈ 𝐶) → {𝑓 ∣ 𝑓:𝐵⟶𝐴} ∈ V) | |
2 | 1 | ancoms 459 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ 𝑓:𝐵⟶𝐴} ∈ V) |
3 | elex 3492 | . . 3 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V) | |
4 | elex 3492 | . . 3 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ∈ V) | |
5 | feq3 6700 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑓:𝑦⟶𝑥 ↔ 𝑓:𝑦⟶𝐴)) | |
6 | 5 | abbidv 2801 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑓 ∣ 𝑓:𝑦⟶𝑥} = {𝑓 ∣ 𝑓:𝑦⟶𝐴}) |
7 | feq2 6699 | . . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑓:𝑦⟶𝐴 ↔ 𝑓:𝐵⟶𝐴)) | |
8 | 7 | abbidv 2801 | . . . . 5 ⊢ (𝑦 = 𝐵 → {𝑓 ∣ 𝑓:𝑦⟶𝐴} = {𝑓 ∣ 𝑓:𝐵⟶𝐴}) |
9 | df-map 8824 | . . . . 5 ⊢ ↑m = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑦⟶𝑥}) | |
10 | 6, 8, 9 | ovmpog 7569 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ {𝑓 ∣ 𝑓:𝐵⟶𝐴} ∈ V) → (𝐴 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴}) |
11 | 10 | 3expia 1121 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝑓 ∣ 𝑓:𝐵⟶𝐴} ∈ V → (𝐴 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴})) |
12 | 3, 4, 11 | syl2an 596 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ({𝑓 ∣ 𝑓:𝐵⟶𝐴} ∈ V → (𝐴 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴})) |
13 | 2, 12 | mpd 15 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {cab 2709 Vcvv 3474 ⟶wf 6539 (class class class)co 7411 ↑m cmap 8822 |
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 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-map 8824 |
This theorem is referenced by: mapval 8834 elmapg 8835 ixpconstg 8902 hashf1lem2 14421 efmndbasabf 18789 symgbasfi 19287 birthdaylem1 26680 birthdaylem2 26681 cnfex 44014 |
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