Theorem infcl 9456

Description: An infimum belongs to its base class (closure law). See also inflb 9457 and infglb 9458. (Contributed by AV, 3-Sep-2020.)

Hypotheses

Ref	Expression
infcl.1	⊢ (𝜑 → 𝑅 Or 𝐴)
infcl.2	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))

Assertion

Ref	Expression
infcl	⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐴)

Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧)

Proof of Theorem infcl

Step	Hyp	Ref	Expression
1		df-inf 9410	. 2 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅)
2		infcl.1	. . . 4 ⊢ (𝜑 → 𝑅 Or 𝐴)
3		cnvso 6293	. . . 4 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴)
4	2, 3	sylib 221	. . 3 ⊢ (𝜑 → ◡𝑅 Or 𝐴)
5		infcl.2	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))
6	2, 5	infcllem 9455	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))
7	4, 6	supcl 9425	. 2 ⊢ (𝜑 → sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴)
8	1, 7	eqeltrid 2869	1 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091   class class class wbr 5111   Or wor 5570  ^◡ccnv 5662  supcsup 9407  infcinf 9408

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-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-po 5571  df-so 5572  df-cnv 5671  df-iota 6496  df-riota 7376  df-sup 9409  df-inf 9410

This theorem is used by:  infssd  9461  infrecl  12212  infxrcl  13376  xrge0infssd  33176  infxrge0lb  33179  infxrge0gelb  33181  omsf  34751  wzel  36351  wsuccl  36354