Theorem mapsn 8838

Description: The value of set exponentiation with a singleton exponent. Theorem 98 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Proof shortened by AV, 17-Jul-2022.)

Hypotheses

Ref	Expression
map0.1	⊢ 𝐴 ∈ V
map0.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
mapsn	⊢ (𝐴 ↑m {𝐵}) = {𝑓 ∣ ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}}

Distinct variable groups:   𝑦,𝑓,𝐴   𝐵,𝑓,𝑦

Proof of Theorem mapsn

Step	Hyp	Ref	Expression
1		map0.1	. 2 ⊢ 𝐴 ∈ V
2		id 22	. . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ V)
3		map0.2	. . . 4 ⊢ 𝐵 ∈ V
4	3	a1i 11	. . 3 ⊢ (𝐴 ∈ V → 𝐵 ∈ V)
5	2, 4	mapsnd 8836	. 2 ⊢ (𝐴 ∈ V → (𝐴 ↑m {𝐵}) = {𝑓 ∣ ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}})
6	1, 5	ax-mp 5	1 ⊢ (𝐴 ↑m {𝐵}) = {𝑓 ∣ ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}}

Syntax hints:   = wceq 1540   ∈ wcel 2109  {cab 2707  ∃wrex 3053  Vcvv 3444  {csn 4585  ⟨cop 4591  (class class class)co 7369   ↑_m cmap 8776

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 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-map 8778

This theorem is referenced by: (None)