Theorem idrefALT 5947

Description: Alternate proof of idref 6918 not relying on definitions related to functions. Two ways to state that a relation is reflexive on a class. (Contributed by FL, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) (Revised by NM, 30-Mar-2016.) (Proof shortened by BJ, 28-Aug-2022.) The "proof modification is discouraged" tag is here only because this is an *ALT result. (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
idrefALT	⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)

Distinct variable groups:   𝑥,𝑅   𝑥,𝐴

Proof of Theorem idrefALT

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfss2 3863	. 2 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅))
2		elrid 5887	. . . . . 6 ⊢ (𝑦 ∈ ( I ↾ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑦 = ⟨𝑥, 𝑥⟩)
3	2	imbi1i 353	. . . . 5 ⊢ ((𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ (∃𝑥 ∈ 𝐴 𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦 ∈ 𝑅))
4		r19.23v 3189	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦 ∈ 𝑅) ↔ (∃𝑥 ∈ 𝐴 𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦 ∈ 𝑅))
5		eleq1 2820	. . . . . . . 8 ⊢ (𝑦 = ⟨𝑥, 𝑥⟩ → (𝑦 ∈ 𝑅 ↔ ⟨𝑥, 𝑥⟩ ∈ 𝑅))
6		df-br 5031	. . . . . . . 8 ⊢ (𝑥𝑅𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ 𝑅)
7	5, 6	bitr4di 292	. . . . . . 7 ⊢ (𝑦 = ⟨𝑥, 𝑥⟩ → (𝑦 ∈ 𝑅 ↔ 𝑥𝑅𝑥))
8	7	pm5.74i 274	. . . . . 6 ⊢ ((𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦 ∈ 𝑅) ↔ (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
9	8	ralbii 3080	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑦 ∈ 𝑅) ↔ ∀𝑥 ∈ 𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
10	3, 4, 9	3bitr2i 302	. . . 4 ⊢ ((𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ ∀𝑥 ∈ 𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
11	10	albii 1826	. . 3 ⊢ (∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
12		ralcom4 3148	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥))
13		opex 5322	. . . . 5 ⊢ ⟨𝑥, 𝑥⟩ ∈ V
14		biidd 265	. . . . 5 ⊢ (𝑦 = ⟨𝑥, 𝑥⟩ → (𝑥𝑅𝑥 ↔ 𝑥𝑅𝑥))
15	13, 14	ceqsalv 3433	. . . 4 ⊢ (∀𝑦(𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥) ↔ 𝑥𝑅𝑥)
16	15	ralbii 3080	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 = ⟨𝑥, 𝑥⟩ → 𝑥𝑅𝑥) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)
17	11, 12, 16	3bitr2i 302	. 2 ⊢ (∀𝑦(𝑦 ∈ ( I ↾ 𝐴) → 𝑦 ∈ 𝑅) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)
18	1, 17	bitri 278	1 ⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ∀wral 3053  ∃wrex 3054   ⊆ wss 3843  ⟨cop 4522   class class class wbr 5030   I cid 5428   ↾ cres 5527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-11 2162  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-id 5429  df-xp 5531  df-rel 5532  df-res 5537

This theorem is referenced by:  idinxpssinxp2  36076  idinxpssinxp3  36077  symrefref3  36301  refsymrels3  36303  elrefsymrels3  36307  dfeqvrels3  36325  refrelsredund3  36370  refrelredund3  36373