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Theorem snmapen 9037
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 ((𝐴𝑉𝐵𝑊) → ({𝐴} ↑m 𝐵) ≈ {𝐴})

Proof of Theorem snmapen
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovexd 7439 . 2 ((𝐴𝑉𝐵𝑊) → ({𝐴} ↑m 𝐵) ∈ V)
2 snex 5424 . . 3 {𝐴} ∈ V
32a1i 11 . 2 ((𝐴𝑉𝐵𝑊) → {𝐴} ∈ V)
4 simpl 482 . . 3 ((𝐴𝑉𝐵𝑊) → 𝐴𝑉)
54a1d 25 . 2 ((𝐴𝑉𝐵𝑊) → (𝑥 ∈ ({𝐴} ↑m 𝐵) → 𝐴𝑉))
62a1i 11 . . . . 5 (𝐴𝑉 → {𝐴} ∈ V)
76anim1ci 615 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐵𝑊 ∧ {𝐴} ∈ V))
8 xpexg 7733 . . . 4 ((𝐵𝑊 ∧ {𝐴} ∈ V) → (𝐵 × {𝐴}) ∈ V)
97, 8syl 17 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐵 × {𝐴}) ∈ V)
109a1d 25 . 2 ((𝐴𝑉𝐵𝑊) → (𝑦 ∈ {𝐴} → (𝐵 × {𝐴}) ∈ V))
11 velsn 4639 . . . . 5 (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
1211a1i 11 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴))
13 elmapg 8832 . . . . . 6 (({𝐴} ∈ V ∧ 𝐵𝑊) → (𝑥 ∈ ({𝐴} ↑m 𝐵) ↔ 𝑥:𝐵⟶{𝐴}))
146, 13sylan 579 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝑥 ∈ ({𝐴} ↑m 𝐵) ↔ 𝑥:𝐵⟶{𝐴}))
15 fconst2g 7199 . . . . . 6 (𝐴𝑉 → (𝑥:𝐵⟶{𝐴} ↔ 𝑥 = (𝐵 × {𝐴})))
1615adantr 480 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝑥:𝐵⟶{𝐴} ↔ 𝑥 = (𝐵 × {𝐴})))
1714, 16bitr2d 280 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝑥 = (𝐵 × {𝐴}) ↔ 𝑥 ∈ ({𝐴} ↑m 𝐵)))
1812, 17anbi12d 630 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝑦 ∈ {𝐴} ∧ 𝑥 = (𝐵 × {𝐴})) ↔ (𝑦 = 𝐴𝑥 ∈ ({𝐴} ↑m 𝐵))))
19 ancom 460 . . 3 ((𝑦 = 𝐴𝑥 ∈ ({𝐴} ↑m 𝐵)) ↔ (𝑥 ∈ ({𝐴} ↑m 𝐵) ∧ 𝑦 = 𝐴))
2018, 19bitr2di 288 . 2 ((𝐴𝑉𝐵𝑊) → ((𝑥 ∈ ({𝐴} ↑m 𝐵) ∧ 𝑦 = 𝐴) ↔ (𝑦 ∈ {𝐴} ∧ 𝑥 = (𝐵 × {𝐴}))))
211, 3, 5, 10, 20en2d 8983 1 ((𝐴𝑉𝐵𝑊) → ({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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