Theorem pgindlem 50777

Description: Lemma for pgind 50779. (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 8031	. . 3 ⊢ (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → (1st ‘𝑥) ∈ 𝒫 𝑧)
2	1	elpwid 4566	. 2 ⊢ (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → (1st ‘𝑥) ⊆ 𝑧)
3		xp2nd 8032	. . 3 ⊢ (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → (2nd ‘𝑥) ∈ 𝒫 𝑧)
4	3	elpwid 4566	. 2 ⊢ (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → (2nd ‘𝑥) ⊆ 𝑧)
5	2, 4	unssd 4138	1 ⊢ (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → ((1st ‘𝑥) ∪ (2nd ‘𝑥)) ⊆ 𝑧)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   ∪ cun 3897   ⊆ wss 3899  𝒫 cpw 4557   × cxp 5649  ‘cfv 6537  1^st c1st 7997  2^nd c2nd 7998

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6493  df-fun 6539  df-fv 6545  df-1st 7999  df-2nd 8000

This theorem is used by:  pgindnf  50778