Theorem supiso 9234

Description: Image of a supremum under an isomorphism. (Contributed by Mario Carneiro, 24-Dec-2016.)

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)))
supiso.4	⊢ (𝜑 → 𝑅 Or 𝐴)

Assertion

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

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

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem supiso

Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		supiso.4	. . 3 ⊢ (𝜑 → 𝑅 Or 𝐴)
2		supiso.1	. . . 4 ⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
3		isoso 7219	. . . 4 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐵))
4	2, 3	syl 17	. . 3 ⊢ (𝜑 → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐵))
5	1, 4	mpbid 231	. 2 ⊢ (𝜑 → 𝑆 Or 𝐵)
6		isof1o 7194	. . . 4 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
7		f1of 6716	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵)
8	2, 6, 7	3syl 18	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
9		supisoex.3	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧)))
10	1, 9	supcl 9217	. . 3 ⊢ (𝜑 → sup(𝐶, 𝐴, 𝑅) ∈ 𝐴)
11	8, 10	ffvelrnd 6962	. 2 ⊢ (𝜑 → (𝐹‘sup(𝐶, 𝐴, 𝑅)) ∈ 𝐵)
12	1, 9	supub 9218	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ 𝐶 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑢))
13	12	ralrimiv 3102	. . . . 5 ⊢ (𝜑 → ∀𝑢 ∈ 𝐶 ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑢)
14	1, 9	suplub 9219	. . . . . . 7 ⊢ (𝜑 → ((𝑢 ∈ 𝐴 ∧ 𝑢𝑅sup(𝐶, 𝐴, 𝑅)) → ∃𝑧 ∈ 𝐶 𝑢𝑅𝑧))
15	14	expd 416	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ 𝐴 → (𝑢𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐶 𝑢𝑅𝑧)))
16	15	ralrimiv 3102	. . . . 5 ⊢ (𝜑 → ∀𝑢 ∈ 𝐴 (𝑢𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐶 𝑢𝑅𝑧))
17		supiso.2	. . . . . . 7 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
18	2, 17	supisolem 9232	. . . . . 6 ⊢ ((𝜑 ∧ sup(𝐶, 𝐴, 𝑅) ∈ 𝐴) → ((∀𝑢 ∈ 𝐶 ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑢 ∧ ∀𝑢 ∈ 𝐴 (𝑢𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐶 𝑢𝑅𝑧)) ↔ (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣))))
19	10, 18	mpdan 684	. . . . 5 ⊢ (𝜑 → ((∀𝑢 ∈ 𝐶 ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑢 ∧ ∀𝑢 ∈ 𝐴 (𝑢𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐶 𝑢𝑅𝑧)) ↔ (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣))))
20	13, 16, 19	mpbi2and 709	. . . 4 ⊢ (𝜑 → (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣)))
21	20	simpld 495	. . 3 ⊢ (𝜑 → ∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤)
22	21	r19.21bi 3134	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐹 “ 𝐶)) → ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤)
23	20	simprd 496	. . . 4 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣))
24	23	r19.21bi 3134	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣))
25	24	impr 455	. 2 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)))) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣)
26	5, 11, 22, 25	eqsupd 9216	1 ⊢ (𝜑 → sup((𝐹 “ 𝐶), 𝐵, 𝑆) = (𝐹‘sup(𝐶, 𝐴, 𝑅)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   ⊆ wss 3887   class class class wbr 5074   Or wor 5502   “ cima 5592  ⟶wf 6429  –1-1-onto→wf1o 6432  ‘cfv 6433   Isom wiso 6434  supcsup 9199

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

This theorem is referenced by:  infiso  9267  infrenegsup  11958