Theorem supiso 9468

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 7356	. . . 4 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐵))
4	2, 3	syl 18	. . 3 ⊢ (𝜑 → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐵))
5	1, 4	mpbid 235	. 2 ⊢ (𝜑 → 𝑆 Or 𝐵)
6		isof1o 7331	. . . 4 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
7		f1of 6824	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵)
8	2, 6, 7	3syl 19	. . 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 9450	. . 3 ⊢ (𝜑 → sup(𝐶, 𝐴, 𝑅) ∈ 𝐴)
11	8, 10	ffvelcdmd 7085	. 2 ⊢ (𝜑 → (𝐹‘sup(𝐶, 𝐴, 𝑅)) ∈ 𝐵)
12	1, 9	supub 9451	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ 𝐶 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑢))
13	12	ralrimiv 3154	. . . . 5 ⊢ (𝜑 → ∀𝑢 ∈ 𝐶 ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑢)
14	1, 9	suplub 9452	. . . . . . 7 ⊢ (𝜑 → ((𝑢 ∈ 𝐴 ∧ 𝑢𝑅sup(𝐶, 𝐴, 𝑅)) → ∃𝑧 ∈ 𝐶 𝑢𝑅𝑧))
15	14	expd 421	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ 𝐴 → (𝑢𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐶 𝑢𝑅𝑧)))
16	15	ralrimiv 3154	. . . . 5 ⊢ (𝜑 → ∀𝑢 ∈ 𝐴 (𝑢𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐶 𝑢𝑅𝑧))
17		supiso.2	. . . . . . 7 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
18	2, 17	supisolem 9466	. . . . . 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 700	. . . . 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 725	. . . 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 500	. . 3 ⊢ (𝜑 → ∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤)
22	21	r19.21bi 3255	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐹 “ 𝐶)) → ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤)
23	20	simprd 501	. . . 4 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣))
24	23	r19.21bi 3255	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣))
25	24	impr 460	. 2 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)))) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣)
26	5, 11, 22, 25	eqsupd 9449	1 ⊢ (𝜑 → sup((𝐹 “ 𝐶), 𝐵, 𝑆) = (𝐹‘sup(𝐶, 𝐴, 𝑅)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087   ⊆ wss 3899   class class class wbr 5103   Or wor 5558   “ cima 5654  ⟶wf 6534  –1-1-onto→wf1o 6537  ‘cfv 6538   Isom wiso 6539  supcsup 9432

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-po 5559  df-so 5560  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-sup 9434

This theorem is used by:  infiso  9502  infrenegsup  12300