Theorem snmapen 8970

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 7388	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} ↑m 𝐵) ∈ V)
2		snex 5378	. . 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 616	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ 𝑊 ∧ {𝐴} ∈ V))
8		xpexg 7690	. . . 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 4595	. . . . 5 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
12	11	a1i 11	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴))
13		elmapg 8773	. . . . . 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 7143	. . . . . 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 632	. . 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 8920	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} ↑m 𝐵) ≈ {𝐴})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3438  {csn 4579   class class class wbr 5095   × cxp 5621  ⟶wf 6482  (class class class)co 7353   ↑_m cmap 8760   ≈ cen 8876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-map 8762  df-en 8880

This theorem is referenced by:  snmapen1  8971