Theorem intimass2 40356

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 40355	. 2 ⊢ (∩ 𝐴 “ 𝐵) ⊆ ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝑥 “ 𝐵)}
2		intima0 40348	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 (𝑥 “ 𝐵) = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝑥 “ 𝐵)}
3	1, 2	sseqtrri 3952	1 ⊢ (∩ 𝐴 “ 𝐵) ⊆ ∩ 𝑥 ∈ 𝐴 (𝑥 “ 𝐵)

Syntax hints:   = wceq 1538  {cab 2776  ∃wrex 3107   ⊆ wss 3881  ∩ cint 4838  ∩ ciin 4882   “ cima 5522

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-iin 4884  df-br 5031  df-opab 5093  df-xp 5525  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532

This theorem is referenced by: (None)