Theorem mapsn 8611

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 8609	. 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 1543   ∈ wcel 2112  {cab 2716  ∃wrex 3065  Vcvv 3423  {csn 4558  ⟨cop 4564  (class class class)co 7252   ↑_m cmap 8550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-sep 5216  ax-nul 5223  ax-pow 5282  ax-pr 5346  ax-un 7563

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3425  df-sbc 3713  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4255  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5479  df-xp 5585  df-rel 5586  df-cnv 5587  df-co 5588  df-dm 5589  df-rn 5590  df-res 5591  df-ima 5592  df-iota 6373  df-fun 6417  df-fn 6418  df-f 6419  df-f1 6420  df-fo 6421  df-f1o 6422  df-fv 6423  df-ov 7255  df-oprab 7256  df-mpo 7257  df-map 8552

This theorem is referenced by: (None)