Theorem supisoti 6423

Description: Image of a supremum under an isomorphism. (Contributed by Jim Kingdon, 26-Nov-2021.)

Hypotheses

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

Assertion

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

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

Allowed substitution hints:   𝜑(𝑦,𝑧)

Proof of Theorem supisoti

Dummy variables 𝑤 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		supisoti.ti	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
2	1	ralrimivva 2443	. . . . . 6 ⊢ (𝜑 → ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
3		supiso.1	. . . . . . 7 ⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
4		isoti 6420	. . . . . . 7 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) ↔ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢))))
5	3, 4	syl 14	. . . . . 6 ⊢ (𝜑 → (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) ↔ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢))))
6	2, 5	mpbid 145	. . . . 5 ⊢ (𝜑 → ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢)))
7	6	r19.21bi 2449	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵) → ∀𝑣 ∈ 𝐵 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢)))
8	7	r19.21bi 2449	. . 3 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢)))
9	8	anasss 391	. 2 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢)))
10		isof1o 5467	. . . 4 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
11		f1of 5146	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵)
12	3, 10, 11	3syl 17	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
13		supisoex.3	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧)))
14	1, 13	supclti 6411	. . 3 ⊢ (𝜑 → sup(𝐶, 𝐴, 𝑅) ∈ 𝐴)
15	12, 14	ffvelrnd 5324	. 2 ⊢ (𝜑 → (𝐹‘sup(𝐶, 𝐴, 𝑅)) ∈ 𝐵)
16	1, 13	supubti 6412	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ 𝐶 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑗))
17	16	ralrimiv 2433	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ 𝐶 ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑗)
18	1, 13	suplubti 6413	. . . . . . 7 ⊢ (𝜑 → ((𝑗 ∈ 𝐴 ∧ 𝑗𝑅sup(𝐶, 𝐴, 𝑅)) → ∃𝑧 ∈ 𝐶 𝑗𝑅𝑧))
19	18	expd 254	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ 𝐴 → (𝑗𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐶 𝑗𝑅𝑧)))
20	19	ralrimiv 2433	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 (𝑗𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐶 𝑗𝑅𝑧))
21		supiso.2	. . . . . . 7 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
22	3, 21	supisolem 6421	. . . . . 6 ⊢ ((𝜑 ∧ sup(𝐶, 𝐴, 𝑅) ∈ 𝐴) → ((∀𝑗 ∈ 𝐶 ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑗 ∧ ∀𝑗 ∈ 𝐴 (𝑗𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐶 𝑗𝑅𝑧)) ↔ (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑘 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑘))))
23	14, 22	mpdan 412	. . . . 5 ⊢ (𝜑 → ((∀𝑗 ∈ 𝐶 ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑗 ∧ ∀𝑗 ∈ 𝐴 (𝑗𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐶 𝑗𝑅𝑧)) ↔ (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑘 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑘))))
24	17, 20, 23	mpbi2and 884	. . . 4 ⊢ (𝜑 → (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑘 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑘)))
25	24	simpld 110	. . 3 ⊢ (𝜑 → ∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤)
26	25	r19.21bi 2449	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐹 “ 𝐶)) → ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤)
27	24	simprd 112	. . . 4 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑘 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑘))
28	27	r19.21bi 2449	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑘 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑘))
29	28	impr 371	. 2 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)))) → ∃𝑘 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑘)
30	9, 15, 26, 29	eqsuptid 6410	1 ⊢ (𝜑 → sup((𝐹 “ 𝐶), 𝐵, 𝑆) = (𝐹‘sup(𝐶, 𝐴, 𝑅)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284   ∈ wcel 1433  ∀wral 2348  ∃wrex 2349   ⊆ wss 2973   class class class wbr 3785   “ cima 4366  ⟶wf 4918  –1-1-onto→wf1o 4921  ‘cfv 4922   Isom wiso 4923  supcsup 6395

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-isom 4931  df-riota 5488  df-sup 6397

This theorem is referenced by:  infisoti  6445  infrenegsupex  8682