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Theorem suplub2 8534
Description: Bidirectional form of suplub 8533. (Contributed by Mario Carneiro, 6-Sep-2014.)
Hypotheses
Ref Expression
supmo.1 (𝜑𝑅 Or 𝐴)
supcl.2 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
suplub2.3 (𝜑𝐵𝐴)
Assertion
Ref Expression
suplub2 ((𝜑𝐶𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ ∃𝑧𝐵 𝐶𝑅𝑧))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝑅,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑧,𝐶
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐶(𝑥,𝑦)

Proof of Theorem suplub2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 supmo.1 . . . 4 (𝜑𝑅 Or 𝐴)
2 supcl.2 . . . 4 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
31, 2suplub 8533 . . 3 (𝜑 → ((𝐶𝐴𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧𝐵 𝐶𝑅𝑧))
43expdimp 452 . 2 ((𝜑𝐶𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝐶𝑅𝑧))
5 breq2 4808 . . . 4 (𝑧 = 𝑤 → (𝐶𝑅𝑧𝐶𝑅𝑤))
65cbvrexv 3311 . . 3 (∃𝑧𝐵 𝐶𝑅𝑧 ↔ ∃𝑤𝐵 𝐶𝑅𝑤)
7 breq2 4808 . . . . . . 7 (sup(𝐵, 𝐴, 𝑅) = 𝑤 → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ 𝐶𝑅𝑤))
87biimprd 238 . . . . . 6 (sup(𝐵, 𝐴, 𝑅) = 𝑤 → (𝐶𝑅𝑤𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
98a1i 11 . . . . 5 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → (sup(𝐵, 𝐴, 𝑅) = 𝑤 → (𝐶𝑅𝑤𝐶𝑅sup(𝐵, 𝐴, 𝑅))))
101ad2antrr 764 . . . . . . 7 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → 𝑅 Or 𝐴)
11 simplr 809 . . . . . . 7 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → 𝐶𝐴)
12 suplub2.3 . . . . . . . . 9 (𝜑𝐵𝐴)
1312adantr 472 . . . . . . . 8 ((𝜑𝐶𝐴) → 𝐵𝐴)
1413sselda 3744 . . . . . . 7 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → 𝑤𝐴)
151, 2supcl 8531 . . . . . . . 8 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
1615ad2antrr 764 . . . . . . 7 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
17 sotr 5209 . . . . . . 7 ((𝑅 Or 𝐴 ∧ (𝐶𝐴𝑤𝐴 ∧ sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)) → ((𝐶𝑅𝑤𝑤𝑅sup(𝐵, 𝐴, 𝑅)) → 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
1810, 11, 14, 16, 17syl13anc 1479 . . . . . 6 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → ((𝐶𝑅𝑤𝑤𝑅sup(𝐵, 𝐴, 𝑅)) → 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
1918expcomd 453 . . . . 5 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → (𝐶𝑅𝑤𝐶𝑅sup(𝐵, 𝐴, 𝑅))))
201, 2supub 8532 . . . . . . . 8 (𝜑 → (𝑤𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
2120adantr 472 . . . . . . 7 ((𝜑𝐶𝐴) → (𝑤𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
2221imp 444 . . . . . 6 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤)
23 sotric 5213 . . . . . . . 8 ((𝑅 Or 𝐴 ∧ (sup(𝐵, 𝐴, 𝑅) ∈ 𝐴𝑤𝐴)) → (sup(𝐵, 𝐴, 𝑅)𝑅𝑤 ↔ ¬ (sup(𝐵, 𝐴, 𝑅) = 𝑤𝑤𝑅sup(𝐵, 𝐴, 𝑅))))
2410, 16, 14, 23syl12anc 1475 . . . . . . 7 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → (sup(𝐵, 𝐴, 𝑅)𝑅𝑤 ↔ ¬ (sup(𝐵, 𝐴, 𝑅) = 𝑤𝑤𝑅sup(𝐵, 𝐴, 𝑅))))
2524con2bid 343 . . . . . 6 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → ((sup(𝐵, 𝐴, 𝑅) = 𝑤𝑤𝑅sup(𝐵, 𝐴, 𝑅)) ↔ ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝑤))
2622, 25mpbird 247 . . . . 5 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → (sup(𝐵, 𝐴, 𝑅) = 𝑤𝑤𝑅sup(𝐵, 𝐴, 𝑅)))
279, 19, 26mpjaod 395 . . . 4 (((𝜑𝐶𝐴) ∧ 𝑤𝐵) → (𝐶𝑅𝑤𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
2827rexlimdva 3169 . . 3 ((𝜑𝐶𝐴) → (∃𝑤𝐵 𝐶𝑅𝑤𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
296, 28syl5bi 232 . 2 ((𝜑𝐶𝐴) → (∃𝑧𝐵 𝐶𝑅𝑧𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
304, 29impbid 202 1 ((𝜑𝐶𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ ∃𝑧𝐵 𝐶𝑅𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  wss 3715   class class class wbr 4804   Or wor 5186  supcsup 8513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-po 5187  df-so 5188  df-iota 6012  df-riota 6775  df-sup 8515
This theorem is referenced by:  infglbb  8564  suprlub  11199  supxrlub  12368
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