Theorem iuneq2 4980

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 4979	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶)
2		ss2iun 4979	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
3	1, 2	anim12i 624	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵) → (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶 ∧ ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵))
4		eqss 3960	. . . 4 ⊢ (𝐵 = 𝐶 ↔ (𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵))
5	4	ralbii 3117	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 ↔ ∀𝑥 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵))
6		r19.26 3131	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ↔ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵))
7	5, 6	bitri 278	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 ↔ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵))
8		eqss 3960	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶 ↔ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶 ∧ ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵))
9	3, 7, 8	3imtr4i 295	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567  ∀wral 3085   ⊆ wss 3913  ∪ ciun 4960

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-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-v 3465  df-ss 3930  df-iun 4962

This theorem is referenced by:  iuneq2i  4982  iuneq2dv  4985  iunxdif3  5065  oa0r  8525  om0r  8526  om1r  8530  oe1m  8532  oaass  8548  oarec  8549  omass  8567  oeoalem  8584  oeoelem  8586  cardiun  9970  kmlem11  10146  iuncld  23173  comppfsc  23660  istotbnd3  38347  sstotbnd  38351  heibor  38397  iuneq12f  38739  cnvtrclfv  44379  iuneq2df  45696