Theorem infcl 9416

Description: An infimum belongs to its base class (closure law). See also inflb 9417 and infglb 9418. (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 9370	. 2 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅)
2		infcl.1	. . . 4 ⊢ (𝜑 → 𝑅 Or 𝐴)
3		cnvso 6249	. . . 4 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴)
4	2, 3	sylib 218	. . 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 9415	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))
7	4, 6	supcl 9385	. 2 ⊢ (𝜑 → sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴)
8	1, 7	eqeltrid 2832	1 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   class class class wbr 5102   Or wor 5538  ^◡ccnv 5630  supcsup 9367  infcinf 9368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-po 5539  df-so 5540  df-cnv 5639  df-iota 6452  df-riota 7326  df-sup 9369  df-inf 9370

This theorem is referenced by:  infssd  9421  infrecl  12141  infxrcl  13270  xrge0infssd  32657  infxrge0lb  32660  infxrge0gelb  32662  omsf  34260  wzel  35785  wsuccl  35788