Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 7863 | . . 3 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝐴 ∈ 𝐶) → {𝑓 ∣ 𝑓:𝐵⟶𝐴} ∈ V) | |
2 | 1 | ancoms 469 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ 𝑓:𝐵⟶𝐴} ∈ V) |
3 | elex 3212 | . . 3 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V) | |
4 | elex 3212 | . . 3 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ∈ V) | |
5 | feq3 6028 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑓:𝑦⟶𝑥 ↔ 𝑓:𝑦⟶𝐴)) | |
6 | 5 | abbidv 2741 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑓 ∣ 𝑓:𝑦⟶𝑥} = {𝑓 ∣ 𝑓:𝑦⟶𝐴}) |
7 | feq2 6027 | . . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑓:𝑦⟶𝐴 ↔ 𝑓:𝐵⟶𝐴)) | |
8 | 7 | abbidv 2741 | . . . . 5 ⊢ (𝑦 = 𝐵 → {𝑓 ∣ 𝑓:𝑦⟶𝐴} = {𝑓 ∣ 𝑓:𝐵⟶𝐴}) |
9 | df-map 7859 | . . . . 5 ⊢ ↑𝑚 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑦⟶𝑥}) | |
10 | 6, 8, 9 | ovmpt2g 6795 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ {𝑓 ∣ 𝑓:𝐵⟶𝐴} ∈ V) → (𝐴 ↑𝑚 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴}) |
11 | 10 | 3expia 1267 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝑓 ∣ 𝑓:𝐵⟶𝐴} ∈ V → (𝐴 ↑𝑚 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴})) |
12 | 3, 4, 11 | syl2an 494 | . 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 384 = wceq 1483 ∈ wcel 1990 {cab 2608 Vcvv 3200 ⟶wf 5884 (class class class)co 6650 ↑𝑚 cmap 7857 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-map 7859 |
This theorem is referenced by: mapval 7869 elmapg 7870 ixpconstg 7917 hashf1lem2 13240 symgbasfi 17806 birthdaylem1 24678 birthdaylem2 24679 cnfex 39187 |
Copyright terms: Public domain | W3C validator |