Theorem infisoti 7133

Description: Image of an infimum under an isomorphism. (Contributed by Jim Kingdon, 19-Dec-2021.)

Hypotheses

Ref	Expression
infisoti.1	⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
infisoti.2	⊢ (𝜑 → 𝐶 ⊆ 𝐴)
infisoti.3	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐶 𝑧𝑅𝑦)))
infisoti.ti	⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))

Assertion

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

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

Proof of Theorem infisoti

Step	Hyp	Ref	Expression
1		infisoti.1	. . . 4 ⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
2		isocnv2 5880	. . . 4 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐹 Isom ◡𝑅, ◡𝑆(𝐴, 𝐵))
3	1, 2	sylib 122	. . 3 ⊢ (𝜑 → 𝐹 Isom ◡𝑅, ◡𝑆(𝐴, 𝐵))
4		infisoti.2	. . 3 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
5		infisoti.3	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐶 𝑧𝑅𝑦)))
6	5	cnvinfex 7119	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦◡𝑅𝑧)))
7		infisoti.ti	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
8	7	cnvti 7120	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢◡𝑅𝑣 ∧ ¬ 𝑣◡𝑅𝑢)))
9	3, 4, 6, 8	supisoti 7111	. 2 ⊢ (𝜑 → sup((𝐹 “ 𝐶), 𝐵, ◡𝑆) = (𝐹‘sup(𝐶, 𝐴, ◡𝑅)))
10		df-inf 7086	. 2 ⊢ inf((𝐹 “ 𝐶), 𝐵, 𝑆) = sup((𝐹 “ 𝐶), 𝐵, ◡𝑆)
11		df-inf 7086	. . 3 ⊢ inf(𝐶, 𝐴, 𝑅) = sup(𝐶, 𝐴, ◡𝑅)
12	11	fveq2i 5578	. 2 ⊢ (𝐹‘inf(𝐶, 𝐴, 𝑅)) = (𝐹‘sup(𝐶, 𝐴, ◡𝑅))
13	9, 10, 12	3eqtr4g 2262	1 ⊢ (𝜑 → inf((𝐹 “ 𝐶), 𝐵, 𝑆) = (𝐹‘inf(𝐶, 𝐴, 𝑅)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1372   ∈ wcel 2175  ∀wral 2483  ∃wrex 2484   ⊆ wss 3165   class class class wbr 4043  ^◡ccnv 4673   “ cima 4677  ‘cfv 5270   Isom wiso 5271  supcsup 7083  infcinf 7084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-isom 5279  df-riota 5898  df-sup 7085  df-inf 7086

This theorem is referenced by: (None)