Theorem intimass2 44364

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 44363	. 2 ⊢ (∩ 𝐴 “ 𝐵) ⊆ ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝑥 “ 𝐵)}
2		intima0 44357	. 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 1570  {cab 2741  ∃wrex 3089   ⊆ wss 3906  ∩ cint 4913  ∩ ciin 4958   “ cima 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iin 4960  df-br 5111  df-opab 5175  df-xp 5669  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676

This theorem is referenced by: (None)