Theorem snmapen 9054

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.)

Assertion

Ref	Expression
snmapen	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} ↑m 𝐵) ≈ {𝐴})

Proof of Theorem snmapen

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovexd 7449	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} ↑m 𝐵) ∈ V)
2		snex 5427	. . 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 7746	. . . 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 4640	. . . . 5 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
12	11	a1i 11	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴))
13		elmapg 8849	. . . . . 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 7209	. . . . . 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 9000	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} ↑m 𝐵) ≈ {𝐴})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  Vcvv 3469  {csn 4624   class class class wbr 5142   × cxp 5670  ⟶wf 6538  (class class class)co 7414   ↑_m cmap 8836   ≈ cen 8952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-map 8838  df-en 8956

This theorem is referenced by:  snmapen1  9055