Theorem ttciun 37120

Description: Distribute indexed union through a transitive closure. (Contributed by Matthew House, 6-Apr-2026.)

Assertion

Ref	Expression
ttciun	⊢ TC+ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 TC+ 𝐵

Proof of Theorem ttciun

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iunxiun 5061	. . . 4 ⊢ ∪ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵TC+ 𝑦 = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 TC+ 𝑦
2	1	uneq1i 4114	. . 3 ⊢ (∪ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵TC+ 𝑦 ∪ ∪ 𝑥 ∈ 𝐴 𝐵) = (∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 TC+ 𝑦 ∪ ∪ 𝑥 ∈ 𝐴 𝐵)
3		iunun 5057	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 (∪ 𝑦 ∈ 𝐵 TC+ 𝑦 ∪ 𝐵) = (∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 TC+ 𝑦 ∪ ∪ 𝑥 ∈ 𝐴 𝐵)
4	2, 3	eqtr4i 2788	. 2 ⊢ (∪ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵TC+ 𝑦 ∪ ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (∪ 𝑦 ∈ 𝐵 TC+ 𝑦 ∪ 𝐵)
5		ttciunun 37117	. 2 ⊢ TC+ ∪ 𝑥 ∈ 𝐴 𝐵 = (∪ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵TC+ 𝑦 ∪ ∪ 𝑥 ∈ 𝐴 𝐵)
6		ttciunun 37117	. . . 4 ⊢ TC+ 𝐵 = (∪ 𝑦 ∈ 𝐵 TC+ 𝑦 ∪ 𝐵)
7	6	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐴 → TC+ 𝐵 = (∪ 𝑦 ∈ 𝐵 TC+ 𝑦 ∪ 𝐵))
8	7	iuneq2i 4976	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 TC+ 𝐵 = ∪ 𝑥 ∈ 𝐴 (∪ 𝑦 ∈ 𝐵 TC+ 𝑦 ∪ 𝐵)
9	4, 5, 8	3eqtr4i 2795	1 ⊢ TC+ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 TC+ 𝐵

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145   ∪ cun 3900  ∪ ciun 4954  TC+ cttc 37092

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-om 7866  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-ttc 37093

This theorem is used by: (None)