Theorem intimass2 43628

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
intimass2	⊢ (∩ 𝐴 “ 𝐵) ⊆ ∩ 𝑥 ∈ 𝐴 (𝑥 “ 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem intimass2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		intimass 43627	. 2 ⊢ (∩ 𝐴 “ 𝐵) ⊆ ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝑥 “ 𝐵)}
2		intima0 43621	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 (𝑥 “ 𝐵) = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝑥 “ 𝐵)}
3	1, 2	sseqtrri 3987	1 ⊢ (∩ 𝐴 “ 𝐵) ⊆ ∩ 𝑥 ∈ 𝐴 (𝑥 “ 𝐵)

Syntax hints:   = wceq 1540  {cab 2707  ∃wrex 3053   ⊆ wss 3905  ∩ cint 4899  ∩ ciin 4945   “ cima 5626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iin 4947  df-br 5096  df-opab 5158  df-xp 5629  df-cnv 5631  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636

This theorem is referenced by: (None)