Theorem iuneq2 5034

Description: Equality theorem for indexed union. (Contributed by NM, 22-Oct-2003.)

Assertion

Ref	Expression
iuneq2	⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶)

Proof of Theorem iuneq2

Step	Hyp	Ref	Expression
1		ss2iun 5033	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶)
2		ss2iun 5033	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
3	1, 2	anim12i 612	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵) → (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶 ∧ ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵))
4		eqss 4024	. . . 4 ⊢ (𝐵 = 𝐶 ↔ (𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵))
5	4	ralbii 3099	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 ↔ ∀𝑥 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵))
6		r19.26 3117	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ↔ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵))
7	5, 6	bitri 275	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 ↔ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵))
8		eqss 4024	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶 ↔ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶 ∧ ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵))
9	3, 7, 8	3imtr4i 292	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537  ∀wral 3067   ⊆ wss 3976  ∪ ciun 5015

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-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-v 3490  df-ss 3993  df-iun 5017

This theorem is referenced by:  iuneq2i  5036  iuneq2dv  5039  iunxdif3  5118  oa0r  8594  om0r  8595  om1r  8599  oe1m  8601  oaass  8617  oarec  8618  omass  8636  oeoalem  8652  oeoelem  8654  cardiun  10051  kmlem11  10230  iuncld  23074  comppfsc  23561  istotbnd3  37731  sstotbnd  37735  heibor  37781  iuneq12f  38123  cnvtrclfv  43686  iuneq2df  44948