Theorem rntrclfvOAI 43137

Description: The range of the transitive closure is equal to the range of the relation. (Contributed by OpenAI, 7-Jul-2020.)

Assertion

Ref	Expression
rntrclfvOAI	⊢ (𝑅 ∈ 𝑉 → ran (t+‘𝑅) = ran 𝑅)

Proof of Theorem rntrclfvOAI

Step	Hyp	Ref	Expression
1		trclfvub 14960	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
2		rnss 5888	. . . 4 ⊢ ((t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) → ran (t+‘𝑅) ⊆ ran (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
3	1, 2	syl 17	. . 3 ⊢ (𝑅 ∈ 𝑉 → ran (t+‘𝑅) ⊆ ran (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
4		rnun 6103	. . . . 5 ⊢ ran (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (ran 𝑅 ∪ ran (dom 𝑅 × ran 𝑅))
5	4	a1i 11	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ran (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (ran 𝑅 ∪ ran (dom 𝑅 × ran 𝑅)))
6		rnxpss 6130	. . . . 5 ⊢ ran (dom 𝑅 × ran 𝑅) ⊆ ran 𝑅
7		ssequn2 4130	. . . . 5 ⊢ (ran (dom 𝑅 × ran 𝑅) ⊆ ran 𝑅 ↔ (ran 𝑅 ∪ ran (dom 𝑅 × ran 𝑅)) = ran 𝑅)
8	6, 7	mpbi 230	. . . 4 ⊢ (ran 𝑅 ∪ ran (dom 𝑅 × ran 𝑅)) = ran 𝑅
9	5, 8	eqtrdi 2788	. . 3 ⊢ (𝑅 ∈ 𝑉 → ran (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = ran 𝑅)
10	3, 9	sseqtrd 3959	. 2 ⊢ (𝑅 ∈ 𝑉 → ran (t+‘𝑅) ⊆ ran 𝑅)
11		trclfvlb 14961	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ⊆ (t+‘𝑅))
12		rnss 5888	. . 3 ⊢ (𝑅 ⊆ (t+‘𝑅) → ran 𝑅 ⊆ ran (t+‘𝑅))
13	11, 12	syl 17	. 2 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ⊆ ran (t+‘𝑅))
14	10, 13	eqssd 3940	1 ⊢ (𝑅 ∈ 𝑉 → ran (t+‘𝑅) = ran 𝑅)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ∪ cun 3888   ⊆ wss 3890   × cxp 5622  dom cdm 5624  ran crn 5625  ‘cfv 6492  t+ctcl 14938

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-sep 5231  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  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-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-iota 6448  df-fun 6494  df-fv 6500  df-trcl 14940

This theorem is referenced by: (None)