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Mirrors > Home > MPE Home > Th. List > snmapen | Structured version Visualization version GIF version |
Description: Set exponentiation: a singleton to any set is equinumerous to that singleton. (Contributed by NM, 17-Dec-2003.) (Revised by AV, 17-Jul-2022.) |
Ref | Expression |
---|---|
snmapen | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} ↑m 𝐵) ≈ {𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovexd 7439 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} ↑m 𝐵) ∈ V) | |
2 | snex 5424 | . . 3 ⊢ {𝐴} ∈ V | |
3 | 2 | a1i 11 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴} ∈ V) |
4 | simpl 482 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ 𝑉) | |
5 | 4 | a1d 25 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ ({𝐴} ↑m 𝐵) → 𝐴 ∈ 𝑉)) |
6 | 2 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V) |
7 | 6 | anim1ci 615 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ 𝑊 ∧ {𝐴} ∈ V)) |
8 | xpexg 7733 | . . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ {𝐴} ∈ V) → (𝐵 × {𝐴}) ∈ V) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 × {𝐴}) ∈ V) |
10 | 9 | a1d 25 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑦 ∈ {𝐴} → (𝐵 × {𝐴}) ∈ V)) |
11 | velsn 4639 | . . . . 5 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴) | |
12 | 11 | a1i 11 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)) |
13 | elmapg 8832 | . . . . . 6 ⊢ (({𝐴} ∈ V ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ ({𝐴} ↑m 𝐵) ↔ 𝑥:𝐵⟶{𝐴})) | |
14 | 6, 13 | sylan 579 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ ({𝐴} ↑m 𝐵) ↔ 𝑥:𝐵⟶{𝐴})) |
15 | fconst2g 7199 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑥:𝐵⟶{𝐴} ↔ 𝑥 = (𝐵 × {𝐴}))) | |
16 | 15 | adantr 480 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥:𝐵⟶{𝐴} ↔ 𝑥 = (𝐵 × {𝐴}))) |
17 | 14, 16 | bitr2d 280 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 = (𝐵 × {𝐴}) ↔ 𝑥 ∈ ({𝐴} ↑m 𝐵))) |
18 | 12, 17 | anbi12d 630 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝑦 ∈ {𝐴} ∧ 𝑥 = (𝐵 × {𝐴})) ↔ (𝑦 = 𝐴 ∧ 𝑥 ∈ ({𝐴} ↑m 𝐵)))) |
19 | ancom 460 | . . 3 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 ∈ ({𝐴} ↑m 𝐵)) ↔ (𝑥 ∈ ({𝐴} ↑m 𝐵) ∧ 𝑦 = 𝐴)) | |
20 | 18, 19 | bitr2di 288 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝑥 ∈ ({𝐴} ↑m 𝐵) ∧ 𝑦 = 𝐴) ↔ (𝑦 ∈ {𝐴} ∧ 𝑥 = (𝐵 × {𝐴})))) |
21 | 1, 3, 5, 10, 20 | en2d 8983 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} ↑m 𝐵) ≈ {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3468 {csn 4623 class class class wbr 5141 × cxp 5667 ⟶wf 6532 (class class class)co 7404 ↑m cmap 8819 ≈ cen 8935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-map 8821 df-en 8939 |
This theorem is referenced by: snmapen1 9038 |
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