Theorem ttciun 36702

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 5040	. . . 4 ⊢ ∪ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵TC+ 𝑦 = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 TC+ 𝑦
2	1	uneq1i 4105	. . 3 ⊢ (∪ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵TC+ 𝑦 ∪ ∪ 𝑥 ∈ 𝐴 𝐵) = (∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 TC+ 𝑦 ∪ ∪ 𝑥 ∈ 𝐴 𝐵)
3		iunun 5036	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 (∪ 𝑦 ∈ 𝐵 TC+ 𝑦 ∪ 𝐵) = (∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 TC+ 𝑦 ∪ ∪ 𝑥 ∈ 𝐴 𝐵)
4	2, 3	eqtr4i 2763	. 2 ⊢ (∪ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵TC+ 𝑦 ∪ ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (∪ 𝑦 ∈ 𝐵 TC+ 𝑦 ∪ 𝐵)
5		ttciunun 36699	. 2 ⊢ TC+ ∪ 𝑥 ∈ 𝐴 𝐵 = (∪ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵TC+ 𝑦 ∪ ∪ 𝑥 ∈ 𝐴 𝐵)
6		ttciunun 36699	. . . 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 4956	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 TC+ 𝐵 = ∪ 𝑥 ∈ 𝐴 (∪ 𝑦 ∈ 𝐵 TC+ 𝑦 ∪ 𝐵)
9	4, 5, 8	3eqtr4i 2770	1 ⊢ TC+ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 TC+ 𝐵

Syntax hints:   = wceq 1542   ∈ wcel 2114   ∪ cun 3888  ∪ ciun 4934  TC+ cttc 36674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-ttc 36675

This theorem is referenced by: (None)