Theorem supisoti 7008

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 2559	. . . . . 6 ⊢ (𝜑 → ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
3		supiso.1	. . . . . . 7 ⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
4		isoti 7005	. . . . . . 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 147	. . . . 5 ⊢ (𝜑 → ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢)))
7	6	r19.21bi 2565	. . . 4 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵) → ∀𝑣 ∈ 𝐵 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢)))
8	7	r19.21bi 2565	. . 3 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐵) ∧ 𝑣 ∈ 𝐵) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢)))
9	8	anasss 399	. 2 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢)))
10		isof1o 5807	. . . 4 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
11		f1of 5461	. . . 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 6996	. . 3 ⊢ (𝜑 → sup(𝐶, 𝐴, 𝑅) ∈ 𝐴)
15	12, 14	ffvelcdmd 5652	. 2 ⊢ (𝜑 → (𝐹‘sup(𝐶, 𝐴, 𝑅)) ∈ 𝐵)
16	1, 13	supubti 6997	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ 𝐶 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑗))
17	16	ralrimiv 2549	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ 𝐶 ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑗)
18	1, 13	suplubti 6998	. . . . . . 7 ⊢ (𝜑 → ((𝑗 ∈ 𝐴 ∧ 𝑗𝑅sup(𝐶, 𝐴, 𝑅)) → ∃𝑧 ∈ 𝐶 𝑗𝑅𝑧))
19	18	expd 258	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ 𝐴 → (𝑗𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐶 𝑗𝑅𝑧)))
20	19	ralrimiv 2549	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 (𝑗𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐶 𝑗𝑅𝑧))
21		supiso.2	. . . . . . 7 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
22	3, 21	supisolem 7006	. . . . . 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 421	. . . . 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 943	. . . 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 112	. . 3 ⊢ (𝜑 → ∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤)
26	25	r19.21bi 2565	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐹 “ 𝐶)) → ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤)
27	24	simprd 114	. . . 4 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑘 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑘))
28	27	r19.21bi 2565	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑘 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑘))
29	28	impr 379	. 2 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)))) → ∃𝑘 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑘)
30	9, 15, 26, 29	eqsuptid 6995	1 ⊢ (𝜑 → sup((𝐹 “ 𝐶), 𝐵, 𝑆) = (𝐹‘sup(𝐶, 𝐴, 𝑅)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456   ⊆ wss 3129   class class class wbr 4003   “ cima 4629  ⟶wf 5212  –1-1-onto→wf1o 5215  ‘cfv 5216   Isom wiso 5217  supcsup 6980

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-isom 5225  df-riota 5830  df-sup 6982

This theorem is referenced by:  infisoti  7030  infrenegsupex  9593  infxrnegsupex  11270