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Theorem snmapen 8582
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 7183 . 2 ((𝐴𝑉𝐵𝑊) → ({𝐴} ↑m 𝐵) ∈ V)
2 snex 5322 . . 3 {𝐴} ∈ V
32a1i 11 . 2 ((𝐴𝑉𝐵𝑊) → {𝐴} ∈ V)
4 elex 3511 . . . 4 (𝐴𝑉𝐴 ∈ V)
54adantr 483 . . 3 ((𝐴𝑉𝐵𝑊) → 𝐴 ∈ V)
65a1d 25 . 2 ((𝐴𝑉𝐵𝑊) → (𝑥 ∈ ({𝐴} ↑m 𝐵) → 𝐴 ∈ V))
72a1i 11 . . . . 5 (𝐴𝑉 → {𝐴} ∈ V)
87anim1ci 617 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐵𝑊 ∧ {𝐴} ∈ V))
9 xpexg 7465 . . . 4 ((𝐵𝑊 ∧ {𝐴} ∈ V) → (𝐵 × {𝐴}) ∈ V)
108, 9syl 17 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐵 × {𝐴}) ∈ V)
1110a1d 25 . 2 ((𝐴𝑉𝐵𝑊) → (𝑦 ∈ {𝐴} → (𝐵 × {𝐴}) ∈ V))
12 velsn 4575 . . . . 5 (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
1312a1i 11 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴))
14 elmapg 8411 . . . . . 6 (({𝐴} ∈ V ∧ 𝐵𝑊) → (𝑥 ∈ ({𝐴} ↑m 𝐵) ↔ 𝑥:𝐵⟶{𝐴}))
157, 14sylan 582 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝑥 ∈ ({𝐴} ↑m 𝐵) ↔ 𝑥:𝐵⟶{𝐴}))
16 fconst2g 6958 . . . . . 6 (𝐴𝑉 → (𝑥:𝐵⟶{𝐴} ↔ 𝑥 = (𝐵 × {𝐴})))
1716adantr 483 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝑥:𝐵⟶{𝐴} ↔ 𝑥 = (𝐵 × {𝐴})))
1815, 17bitr2d 282 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝑥 = (𝐵 × {𝐴}) ↔ 𝑥 ∈ ({𝐴} ↑m 𝐵)))
1913, 18anbi12d 632 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝑦 ∈ {𝐴} ∧ 𝑥 = (𝐵 × {𝐴})) ↔ (𝑦 = 𝐴𝑥 ∈ ({𝐴} ↑m 𝐵))))
20 ancom 463 . . 3 ((𝑦 = 𝐴𝑥 ∈ ({𝐴} ↑m 𝐵)) ↔ (𝑥 ∈ ({𝐴} ↑m 𝐵) ∧ 𝑦 = 𝐴))
2119, 20syl6rbb 290 . 2 ((𝐴𝑉𝐵𝑊) → ((𝑥 ∈ ({𝐴} ↑m 𝐵) ∧ 𝑦 = 𝐴) ↔ (𝑦 ∈ {𝐴} ∧ 𝑥 = (𝐵 × {𝐴}))))
221, 3, 6, 11, 21en2d 8537 1 ((𝐴𝑉𝐵𝑊) → ({𝐴} ↑m 𝐵) ≈ {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1531  wcel 2108  Vcvv 3493  {csn 4559   class class class wbr 5057   × cxp 5546  wf 6344  (class class class)co 7148  m cmap 8398  cen 8498
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 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-map 8400  df-en 8502
This theorem is referenced by:  snmapen1  8583
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