Theorem intimag 44006

Description: Requirement for the image under the intersection of relations to equal the intersection of the images of those relations. (Contributed by RP, 13-Apr-2020.)

Assertion

Ref	Expression
intimag	⊢ (∀𝑦(∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎) → (∩ 𝐴 “ 𝐵) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)})

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

Proof of Theorem intimag

Step	Hyp	Ref	Expression
1		r19.12 3287	. . . . 5 ⊢ (∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎 → ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
2		id 22	. . . . 5 ⊢ ((∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎) → (∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
3	1, 2	impbid2 226	. . . 4 ⊢ ((∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎) → (∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎 ↔ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
4		elimaint 43999	. . . 4 ⊢ (𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
5		elintima 44003	. . . 4 ⊢ (𝑦 ∈ ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)} ↔ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
6	3, 4, 5	3bitr4g 314	. . 3 ⊢ ((∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎) → (𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ 𝑦 ∈ ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)}))
7	6	alimi 1813	. 2 ⊢ (∀𝑦(∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎) → ∀𝑦(𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ 𝑦 ∈ ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)}))
8		dfcleq 2730	. 2 ⊢ ((∩ 𝐴 “ 𝐵) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)} ↔ ∀𝑦(𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ 𝑦 ∈ ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)}))
9	7, 8	sylibr 234	1 ⊢ (∀𝑦(∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎) → (∩ 𝐴 “ 𝐵) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)})

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1540   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3062  ⟨cop 4588  ∩ cint 4904   “ cima 5635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-br 5101  df-opab 5163  df-xp 5638  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645

This theorem is referenced by:  intimasn  44007