Theorem ttciun 36879

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 5055	. . . 4 ⊢ ∪ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵TC+ 𝑦 = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 TC+ 𝑦
2	1	uneq1i 4118	. . 3 ⊢ (∪ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵TC+ 𝑦 ∪ ∪ 𝑥 ∈ 𝐴 𝐵) = (∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 TC+ 𝑦 ∪ ∪ 𝑥 ∈ 𝐴 𝐵)
3		iunun 5051	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 (∪ 𝑦 ∈ 𝐵 TC+ 𝑦 ∪ 𝐵) = (∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 TC+ 𝑦 ∪ ∪ 𝑥 ∈ 𝐴 𝐵)
4	2, 3	eqtr4i 2789	. 2 ⊢ (∪ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵TC+ 𝑦 ∪ ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (∪ 𝑦 ∈ 𝐵 TC+ 𝑦 ∪ 𝐵)
5		ttciunun 36876	. 2 ⊢ TC+ ∪ 𝑥 ∈ 𝐴 𝐵 = (∪ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵TC+ 𝑦 ∪ ∪ 𝑥 ∈ 𝐴 𝐵)
6		ttciunun 36876	. . . 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 4972	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 TC+ 𝐵 = ∪ 𝑥 ∈ 𝐴 (∪ 𝑦 ∈ 𝐵 TC+ 𝑦 ∪ 𝐵)
9	4, 5, 8	3eqtr4i 2796	1 ⊢ TC+ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 TC+ 𝐵

Syntax hints:   = wceq 1561   ∈ wcel 2143   ∪ cun 3903  ∪ ciun 4950  TC+ cttc 36851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pr 5391  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-om 7847  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-ttc 36852

This theorem is referenced by: (None)