Theorem snmapen 8820

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 7307	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} ↑m 𝐵) ∈ V)
2		snex 5358	. . 3 ⊢ {𝐴} ∈ V
3	2	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴} ∈ V)
4		simpl 483	. . 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 616	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ 𝑊 ∧ {𝐴} ∈ V))
8		xpexg 7595	. . . 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 4583	. . . . 5 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
12	11	a1i 11	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴))
13		elmapg 8620	. . . . . 6 ⊢ (({𝐴} ∈ V ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ ({𝐴} ↑m 𝐵) ↔ 𝑥:𝐵⟶{𝐴}))
14	6, 13	sylan 580	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ ({𝐴} ↑m 𝐵) ↔ 𝑥:𝐵⟶{𝐴}))
15		fconst2g 7075	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑥:𝐵⟶{𝐴} ↔ 𝑥 = (𝐵 × {𝐴})))
16	15	adantr 481	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥:𝐵⟶{𝐴} ↔ 𝑥 = (𝐵 × {𝐴})))
17	14, 16	bitr2d 279	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 = (𝐵 × {𝐴}) ↔ 𝑥 ∈ ({𝐴} ↑m 𝐵)))
18	12, 17	anbi12d 631	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝑦 ∈ {𝐴} ∧ 𝑥 = (𝐵 × {𝐴})) ↔ (𝑦 = 𝐴 ∧ 𝑥 ∈ ({𝐴} ↑m 𝐵))))
19		ancom 461	. . 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 8768	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} ↑m 𝐵) ≈ {𝐴})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1542   ∈ wcel 2110  Vcvv 3431  {csn 4567   class class class wbr 5079   × cxp 5588  ⟶wf 6428  (class class class)co 7272   ↑_m cmap 8607   ≈ cen 8722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7583

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-ov 7275  df-oprab 7276  df-mpo 7277  df-map 8609  df-en 8726

This theorem is referenced by:  snmapen1  8821