Theorem supiso 9544

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 7384	. . . 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 232	. 2 ⊢ (𝜑 → 𝑆 Or 𝐵)
6		isof1o 7359	. . . 4 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
7		f1of 6862	. . . 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 9527	. . 3 ⊢ (𝜑 → sup(𝐶, 𝐴, 𝑅) ∈ 𝐴)
11	8, 10	ffvelcdmd 7119	. 2 ⊢ (𝜑 → (𝐹‘sup(𝐶, 𝐴, 𝑅)) ∈ 𝐵)
12	1, 9	supub 9528	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ 𝐶 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑢))
13	12	ralrimiv 3151	. . . . 5 ⊢ (𝜑 → ∀𝑢 ∈ 𝐶 ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑢)
14	1, 9	suplub 9529	. . . . . . 7 ⊢ (𝜑 → ((𝑢 ∈ 𝐴 ∧ 𝑢𝑅sup(𝐶, 𝐴, 𝑅)) → ∃𝑧 ∈ 𝐶 𝑢𝑅𝑧))
15	14	expd 415	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ 𝐴 → (𝑢𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐶 𝑢𝑅𝑧)))
16	15	ralrimiv 3151	. . . . 5 ⊢ (𝜑 → ∀𝑢 ∈ 𝐴 (𝑢𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐶 𝑢𝑅𝑧))
17		supiso.2	. . . . . . 7 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
18	2, 17	supisolem 9542	. . . . . 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 686	. . . . 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 711	. . . 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 494	. . 3 ⊢ (𝜑 → ∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤)
22	21	r19.21bi 3257	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐹 “ 𝐶)) → ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤)
23	20	simprd 495	. . . 4 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣))
24	23	r19.21bi 3257	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣))
25	24	impr 454	. 2 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)))) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣)
26	5, 11, 22, 25	eqsupd 9526	1 ⊢ (𝜑 → sup((𝐹 “ 𝐶), 𝐵, 𝑆) = (𝐹‘sup(𝐶, 𝐴, 𝑅)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   ⊆ wss 3976   class class class wbr 5166   Or wor 5606   “ cima 5703  ⟶wf 6569  –1-1-onto→wf1o 6572  ‘cfv 6573   Isom wiso 6574  supcsup 9509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-po 5607  df-so 5608  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-sup 9511

This theorem is referenced by:  infiso  9577  infrenegsup  12278