Theorem pgindlem 48807

Description: Lemma for pgind 48809. (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 8062	. . 3 ⊢ (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → (1st ‘𝑥) ∈ 𝒫 𝑧)
2	1	elpwid 4631	. 2 ⊢ (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → (1st ‘𝑥) ⊆ 𝑧)
3		xp2nd 8063	. . 3 ⊢ (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → (2nd ‘𝑥) ∈ 𝒫 𝑧)
4	3	elpwid 4631	. 2 ⊢ (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → (2nd ‘𝑥) ⊆ 𝑧)
5	2, 4	unssd 4215	1 ⊢ (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → ((1st ‘𝑥) ∪ (2nd ‘𝑥)) ⊆ 𝑧)

Syntax hints:   → wi 4   ∈ wcel 2108   ∪ cun 3974   ⊆ wss 3976  𝒫 cpw 4622   × cxp 5698  ‘cfv 6573  1^st c1st 8028  2^nd c2nd 8029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fv 6581  df-1st 8030  df-2nd 8031

This theorem is referenced by:  pgindnf  48808