Theorem infiso 9267

Description: Image of an infimum under an isomorphism. (Contributed by AV, 4-Sep-2020.)

Hypotheses

Ref	Expression
infiso.1	⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
infiso.2	⊢ (𝜑 → 𝐶 ⊆ 𝐴)
infiso.3	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐶 𝑧𝑅𝑦)))
infiso.4	⊢ (𝜑 → 𝑅 Or 𝐴)

Assertion

Ref	Expression
infiso	⊢ (𝜑 → inf((𝐹 “ 𝐶), 𝐵, 𝑆) = (𝐹‘inf(𝐶, 𝐴, 𝑅)))

Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧

Proof of Theorem infiso

Step	Hyp	Ref	Expression
1		infiso.1	. . . 4 ⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
2		isocnv2 7202	. . . 4 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐹 Isom ◡𝑅, ◡𝑆(𝐴, 𝐵))
3	1, 2	sylib 217	. . 3 ⊢ (𝜑 → 𝐹 Isom ◡𝑅, ◡𝑆(𝐴, 𝐵))
4		infiso.2	. . 3 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
5		infiso.4	. . . 4 ⊢ (𝜑 → 𝑅 Or 𝐴)
6		infiso.3	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐶 𝑧𝑅𝑦)))
7	5, 6	infcllem 9246	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦◡𝑅𝑧)))
8		cnvso 6191	. . . 4 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴)
9	5, 8	sylib 217	. . 3 ⊢ (𝜑 → ◡𝑅 Or 𝐴)
10	3, 4, 7, 9	supiso 9234	. 2 ⊢ (𝜑 → sup((𝐹 “ 𝐶), 𝐵, ◡𝑆) = (𝐹‘sup(𝐶, 𝐴, ◡𝑅)))
11		df-inf 9202	. 2 ⊢ inf((𝐹 “ 𝐶), 𝐵, 𝑆) = sup((𝐹 “ 𝐶), 𝐵, ◡𝑆)
12		df-inf 9202	. . 3 ⊢ inf(𝐶, 𝐴, 𝑅) = sup(𝐶, 𝐴, ◡𝑅)
13	12	fveq2i 6777	. 2 ⊢ (𝐹‘inf(𝐶, 𝐴, 𝑅)) = (𝐹‘sup(𝐶, 𝐴, ◡𝑅))
14	10, 11, 13	3eqtr4g 2803	1 ⊢ (𝜑 → inf((𝐹 “ 𝐶), 𝐵, 𝑆) = (𝐹‘inf(𝐶, 𝐴, 𝑅)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1539  ∀wral 3064  ∃wrex 3065   ⊆ wss 3887   class class class wbr 5074   Or wor 5502  ^◡ccnv 5588   “ cima 5592  ‘cfv 6433   Isom wiso 6434  supcsup 9199  infcinf 9200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-po 5503  df-so 5504  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-sup 9201  df-inf 9202

This theorem is referenced by: (None)