Theorem intimag 44859

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 3315	. . . . 5 ⊢ (∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎 → ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
2		id 23	. . . . 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 229	. . . 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 44852	. . . 4 ⊢ (𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
5		elintima 44856	. . . 4 ⊢ (𝑦 ∈ ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)} ↔ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
6	3, 4, 5	3bitr4g 317	. . 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 1844	. 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 2756	. 2 ⊢ ((∩ 𝐴 “ 𝐵) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)} ↔ ∀𝑦(𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ 𝑦 ∈ ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)}))
9	7, 8	sylibr 237	1 ⊢ (∀𝑦(∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎) → (∩ 𝐴 “ 𝐵) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090  ⟨cop 4838  ∩ cint 4910   “ cima 5659

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-br 5107  df-opab 5172  df-xp 5662  df-rel 5663  df-cnv 5664  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669

This theorem is used by:  intimasn  44860