Theorem intimasn 44186

Description: Two ways to express the image of a singleton when the relation is an intersection. (Contributed by RP, 13-Apr-2020.)

Assertion

Ref	Expression
intimasn	⊢ (𝐵 ∈ 𝑉 → (∩ 𝐴 “ {𝐵}) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ {𝐵})})

Distinct variable groups:   𝐴,𝑎   𝐵,𝑎,𝑥   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝑉(𝑥,𝑎)

Proof of Theorem intimasn

Dummy variables 𝑦 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-5 1929	. 2 ⊢ (𝐵 ∈ 𝑉 → ∀𝑦 𝐵 ∈ 𝑉)
2		r19.12sn 4678	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (∃𝑏 ∈ {𝐵}∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎 ↔ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ {𝐵}⟨𝑏, 𝑦⟩ ∈ 𝑎))
3	2	biimprd 250	. . 3 ⊢ (𝐵 ∈ 𝑉 → (∀𝑎 ∈ 𝐴 ∃𝑏 ∈ {𝐵}⟨𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏 ∈ {𝐵}∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
4	3	alimi 1830	. 2 ⊢ (∀𝑦 𝐵 ∈ 𝑉 → ∀𝑦(∀𝑎 ∈ 𝐴 ∃𝑏 ∈ {𝐵}⟨𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏 ∈ {𝐵}∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
5		intimag 44185	. 2 ⊢ (∀𝑦(∀𝑎 ∈ 𝐴 ∃𝑏 ∈ {𝐵}⟨𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏 ∈ {𝐵}∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎) → (∩ 𝐴 “ {𝐵}) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ {𝐵})})
6	1, 4, 5	3syl 18	1 ⊢ (𝐵 ∈ 𝑉 → (∩ 𝐴 “ {𝐵}) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ {𝐵})})

Syntax hints:   → wi 4  ∀wal 1557   = wceq 1559   ∈ wcel 2141  {cab 2739  ∀wral 3075  ∃wrex 3085  {csn 4581  ⟨cop 4587  ∩ cint 4904   “ cima 5648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-pr 5389  ax-un 7712

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-br 5100  df-opab 5162  df-xp 5651  df-cnv 5653  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658

This theorem is referenced by:  intimasn2  44187  brtrclfv2  44256