Theorem pgindlem 50371

Description: Lemma for pgind 50373. (Contributed by Emmett Weisz, 27-May-2024.) (New usage is discouraged.)

Assertion

Ref	Expression
pgindlem	⊢ (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → ((1st ‘𝑥) ∪ (2nd ‘𝑥)) ⊆ 𝑧)

Proof of Theorem pgindlem

Step	Hyp	Ref	Expression
1		xp1st 8014	. . 3 ⊢ (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → (1st ‘𝑥) ∈ 𝒫 𝑧)
2	1	elpwid 4573	. 2 ⊢ (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → (1st ‘𝑥) ⊆ 𝑧)
3		xp2nd 8015	. . 3 ⊢ (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → (2nd ‘𝑥) ∈ 𝒫 𝑧)
4	3	elpwid 4573	. 2 ⊢ (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → (2nd ‘𝑥) ⊆ 𝑧)
5	2, 4	unssd 4153	1 ⊢ (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → ((1st ‘𝑥) ∪ (2nd ‘𝑥)) ⊆ 𝑧)

Syntax hints:   → wi 4   ∈ wcel 2149   ∪ cun 3911   ⊆ wss 3913  𝒫 cpw 4564   × cxp 5657  ‘cfv 6533  1^st c1st 7980  2^nd c2nd 7981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6489  df-fun 6535  df-fv 6541  df-1st 7982  df-2nd 7983

This theorem is referenced by:  pgindnf  50372