Theorem intimass 42860

Description: The image under the intersection of relations is a subset of the intersection of the images. (Contributed by RP, 13-Apr-2020.)

Assertion

Ref	Expression
intimass	⊢ (∩ 𝐴 “ 𝐵) ⊆ ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)}

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

Proof of Theorem intimass

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

Step	Hyp	Ref	Expression
1		r19.12 3303	. . 3 ⊢ (∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎 → ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
2		elimaint 42855	. . 3 ⊢ (𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
3		elintima 42859	. . 3 ⊢ (𝑦 ∈ ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)} ↔ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
4	1, 2, 3	3imtr4i 292	. 2 ⊢ (𝑦 ∈ (∩ 𝐴 “ 𝐵) → 𝑦 ∈ ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)})
5	4	ssriv 3978	1 ⊢ (∩ 𝐴 “ 𝐵) ⊆ ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)}

Syntax hints:   = wceq 1533   ∈ wcel 2098  {cab 2701  ∀wral 3053  ∃wrex 3062   ⊆ wss 3940  ⟨cop 4626  ∩ cint 4940   “ cima 5669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-br 5139  df-opab 5201  df-xp 5672  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679

This theorem is referenced by:  intimass2  42861