Theorem istsr 18664

Description: The predicate is a toset. (Contributed by FL, 1-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)

Hypothesis

Ref	Expression
istsr.1	⊢ 𝑋 = dom 𝑅

Assertion

Ref	Expression
istsr	⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (𝑋 × 𝑋) ⊆ (𝑅 ∪ ◡𝑅)))

Proof of Theorem istsr

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmeq 5898	. . . . 5 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
2		istsr.1	. . . . 5 ⊢ 𝑋 = dom 𝑅
3	1, 2	eqtr4di 2819	. . . 4 ⊢ (𝑟 = 𝑅 → dom 𝑟 = 𝑋)
4	3	sqxpeqd 5698	. . 3 ⊢ (𝑟 = 𝑅 → (dom 𝑟 × dom 𝑟) = (𝑋 × 𝑋))
5		id 23	. . . 4 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
6		cnveq 5864	. . . 4 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅)
7	5, 6	uneq12d 4126	. . 3 ⊢ (𝑟 = 𝑅 → (𝑟 ∪ ◡𝑟) = (𝑅 ∪ ◡𝑅))
8	4, 7	sseq12d 3973	. 2 ⊢ (𝑟 = 𝑅 → ((dom 𝑟 × dom 𝑟) ⊆ (𝑟 ∪ ◡𝑟) ↔ (𝑋 × 𝑋) ⊆ (𝑅 ∪ ◡𝑅)))
9		df-tsr 18648	. 2 ⊢ TosetRel = {𝑟 ∈ PosetRel ∣ (dom 𝑟 × dom 𝑟) ⊆ (𝑟 ∪ ◡𝑟)}
10	8, 9	elrab2 3657	1 ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (𝑋 × 𝑋) ⊆ (𝑅 ∪ ◡𝑅)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ∪ cun 3906   ⊆ wss 3908   × cxp 5664  ^◡ccnv 5665  dom cdm 5666  PosetRelcps 18645   TosetRel ctsr 18646

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 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5672  df-cnv 5674  df-dm 5676  df-tsr 18648

This theorem is used by:  istsr2  18665  tsrlemax  18667  tsrps  18668  cnvtsr  18669  letsr  18674  tsrdir  18685