Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fnwe2lem2 Structured version   Visualization version   GIF version

Theorem fnwe2lem2 41364
Description: Lemma for fnwe2 41366. An element which is in a minimal fiber and minimal within its fiber is minimal globally; thus 𝑇 is well-founded. (Contributed by Stefan O'Rear, 19-Jan-2015.)
Hypotheses
Ref Expression
fnwe2.su (𝑧 = (𝐹𝑥) → 𝑆 = 𝑈)
fnwe2.t 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑈𝑦))}
fnwe2.s ((𝜑𝑥𝐴) → 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})
fnwe2.f (𝜑 → (𝐹𝐴):𝐴𝐵)
fnwe2.r (𝜑𝑅 We 𝐵)
fnwe2lem2.a (𝜑𝑎𝐴)
fnwe2lem2.n0 (𝜑𝑎 ≠ ∅)
Assertion
Ref Expression
fnwe2lem2 (𝜑 → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑇𝑏)
Distinct variable groups:   𝑦,𝑈,𝑧,𝑎,𝑏,𝑐   𝑥,𝑆,𝑦,𝑎,𝑏,𝑐   𝑥,𝑅,𝑦,𝑎,𝑏,𝑐   𝜑,𝑥,𝑦,𝑧,𝑐   𝑥,𝐴,𝑦,𝑧,𝑎,𝑏,𝑐   𝑥,𝐹,𝑦,𝑧,𝑎,𝑏,𝑐   𝑇,𝑎,𝑏,𝑐   𝐵,𝑎,𝑏,𝑐   𝜑,𝑏
Allowed substitution hints:   𝜑(𝑎)   𝐵(𝑥,𝑦,𝑧)   𝑅(𝑧)   𝑆(𝑧)   𝑇(𝑥,𝑦,𝑧)   𝑈(𝑥)

Proof of Theorem fnwe2lem2
Dummy variables 𝑑 𝑒 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnwe2.f . . . 4 (𝜑 → (𝐹𝐴):𝐴𝐵)
2 ffun 6671 . . . 4 ((𝐹𝐴):𝐴𝐵 → Fun (𝐹𝐴))
3 vex 3449 . . . . 5 𝑎 ∈ V
43funimaex 6589 . . . 4 (Fun (𝐹𝐴) → ((𝐹𝐴) “ 𝑎) ∈ V)
51, 2, 43syl 18 . . 3 (𝜑 → ((𝐹𝐴) “ 𝑎) ∈ V)
6 fnwe2.r . . . 4 (𝜑𝑅 We 𝐵)
7 wefr 5623 . . . 4 (𝑅 We 𝐵𝑅 Fr 𝐵)
86, 7syl 17 . . 3 (𝜑𝑅 Fr 𝐵)
9 imassrn 6024 . . . 4 ((𝐹𝐴) “ 𝑎) ⊆ ran (𝐹𝐴)
101frnd 6676 . . . 4 (𝜑 → ran (𝐹𝐴) ⊆ 𝐵)
119, 10sstrid 3955 . . 3 (𝜑 → ((𝐹𝐴) “ 𝑎) ⊆ 𝐵)
12 incom 4161 . . . . . 6 (dom (𝐹𝐴) ∩ 𝑎) = (𝑎 ∩ dom (𝐹𝐴))
13 fnwe2lem2.a . . . . . . . 8 (𝜑𝑎𝐴)
141fdmd 6679 . . . . . . . 8 (𝜑 → dom (𝐹𝐴) = 𝐴)
1513, 14sseqtrrd 3985 . . . . . . 7 (𝜑𝑎 ⊆ dom (𝐹𝐴))
16 df-ss 3927 . . . . . . 7 (𝑎 ⊆ dom (𝐹𝐴) ↔ (𝑎 ∩ dom (𝐹𝐴)) = 𝑎)
1715, 16sylib 217 . . . . . 6 (𝜑 → (𝑎 ∩ dom (𝐹𝐴)) = 𝑎)
1812, 17eqtrid 2788 . . . . 5 (𝜑 → (dom (𝐹𝐴) ∩ 𝑎) = 𝑎)
19 fnwe2lem2.n0 . . . . 5 (𝜑𝑎 ≠ ∅)
2018, 19eqnetrd 3011 . . . 4 (𝜑 → (dom (𝐹𝐴) ∩ 𝑎) ≠ ∅)
21 imadisj 6032 . . . . 5 (((𝐹𝐴) “ 𝑎) = ∅ ↔ (dom (𝐹𝐴) ∩ 𝑎) = ∅)
2221necon3bii 2996 . . . 4 (((𝐹𝐴) “ 𝑎) ≠ ∅ ↔ (dom (𝐹𝐴) ∩ 𝑎) ≠ ∅)
2320, 22sylibr 233 . . 3 (𝜑 → ((𝐹𝐴) “ 𝑎) ≠ ∅)
24 fri 5593 . . 3 (((((𝐹𝐴) “ 𝑎) ∈ V ∧ 𝑅 Fr 𝐵) ∧ (((𝐹𝐴) “ 𝑎) ⊆ 𝐵 ∧ ((𝐹𝐴) “ 𝑎) ≠ ∅)) → ∃𝑑 ∈ ((𝐹𝐴) “ 𝑎)∀𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅𝑑)
255, 8, 11, 23, 24syl22anc 837 . 2 (𝜑 → ∃𝑑 ∈ ((𝐹𝐴) “ 𝑎)∀𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅𝑑)
26 df-ima 5646 . . . . . 6 ((𝐹𝐴) “ 𝑎) = ran ((𝐹𝐴) ↾ 𝑎)
2726rexeqi 3312 . . . . 5 (∃𝑑 ∈ ((𝐹𝐴) “ 𝑎)∀𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅𝑑 ↔ ∃𝑑 ∈ ran ((𝐹𝐴) ↾ 𝑎)∀𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅𝑑)
281ffnd 6669 . . . . . . 7 (𝜑 → (𝐹𝐴) Fn 𝐴)
29 fnssres 6624 . . . . . . 7 (((𝐹𝐴) Fn 𝐴𝑎𝐴) → ((𝐹𝐴) ↾ 𝑎) Fn 𝑎)
3028, 13, 29syl2anc 584 . . . . . 6 (𝜑 → ((𝐹𝐴) ↾ 𝑎) Fn 𝑎)
31 breq2 5109 . . . . . . . . 9 (𝑑 = (((𝐹𝐴) ↾ 𝑎)‘𝑓) → (𝑒𝑅𝑑𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓)))
3231notbid 317 . . . . . . . 8 (𝑑 = (((𝐹𝐴) ↾ 𝑎)‘𝑓) → (¬ 𝑒𝑅𝑑 ↔ ¬ 𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓)))
3332ralbidv 3174 . . . . . . 7 (𝑑 = (((𝐹𝐴) ↾ 𝑎)‘𝑓) → (∀𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅𝑑 ↔ ∀𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓)))
3433rexrn 7037 . . . . . 6 (((𝐹𝐴) ↾ 𝑎) Fn 𝑎 → (∃𝑑 ∈ ran ((𝐹𝐴) ↾ 𝑎)∀𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅𝑑 ↔ ∃𝑓𝑎𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓)))
3530, 34syl 17 . . . . 5 (𝜑 → (∃𝑑 ∈ ran ((𝐹𝐴) ↾ 𝑎)∀𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅𝑑 ↔ ∃𝑓𝑎𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓)))
3627, 35bitrid 282 . . . 4 (𝜑 → (∃𝑑 ∈ ((𝐹𝐴) “ 𝑎)∀𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅𝑑 ↔ ∃𝑓𝑎𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓)))
3726raleqi 3311 . . . . . . . 8 (∀𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓) ↔ ∀𝑒 ∈ ran ((𝐹𝐴) ↾ 𝑎) ¬ 𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓))
38 breq1 5108 . . . . . . . . . . 11 (𝑒 = (((𝐹𝐴) ↾ 𝑎)‘𝑑) → (𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓) ↔ (((𝐹𝐴) ↾ 𝑎)‘𝑑)𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓)))
3938notbid 317 . . . . . . . . . 10 (𝑒 = (((𝐹𝐴) ↾ 𝑎)‘𝑑) → (¬ 𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓) ↔ ¬ (((𝐹𝐴) ↾ 𝑎)‘𝑑)𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓)))
4039ralrn 7038 . . . . . . . . 9 (((𝐹𝐴) ↾ 𝑎) Fn 𝑎 → (∀𝑒 ∈ ran ((𝐹𝐴) ↾ 𝑎) ¬ 𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓) ↔ ∀𝑑𝑎 ¬ (((𝐹𝐴) ↾ 𝑎)‘𝑑)𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓)))
4130, 40syl 17 . . . . . . . 8 (𝜑 → (∀𝑒 ∈ ran ((𝐹𝐴) ↾ 𝑎) ¬ 𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓) ↔ ∀𝑑𝑎 ¬ (((𝐹𝐴) ↾ 𝑎)‘𝑑)𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓)))
4237, 41bitrid 282 . . . . . . 7 (𝜑 → (∀𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓) ↔ ∀𝑑𝑎 ¬ (((𝐹𝐴) ↾ 𝑎)‘𝑑)𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓)))
4342adantr 481 . . . . . 6 ((𝜑𝑓𝑎) → (∀𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓) ↔ ∀𝑑𝑎 ¬ (((𝐹𝐴) ↾ 𝑎)‘𝑑)𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓)))
4413resabs1d 5968 . . . . . . . . . . . 12 (𝜑 → ((𝐹𝐴) ↾ 𝑎) = (𝐹𝑎))
4544ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝑓𝑎) ∧ 𝑑𝑎) → ((𝐹𝐴) ↾ 𝑎) = (𝐹𝑎))
4645fveq1d 6844 . . . . . . . . . 10 (((𝜑𝑓𝑎) ∧ 𝑑𝑎) → (((𝐹𝐴) ↾ 𝑎)‘𝑑) = ((𝐹𝑎)‘𝑑))
47 fvres 6861 . . . . . . . . . . 11 (𝑑𝑎 → ((𝐹𝑎)‘𝑑) = (𝐹𝑑))
4847adantl 482 . . . . . . . . . 10 (((𝜑𝑓𝑎) ∧ 𝑑𝑎) → ((𝐹𝑎)‘𝑑) = (𝐹𝑑))
4946, 48eqtrd 2776 . . . . . . . . 9 (((𝜑𝑓𝑎) ∧ 𝑑𝑎) → (((𝐹𝐴) ↾ 𝑎)‘𝑑) = (𝐹𝑑))
5045fveq1d 6844 . . . . . . . . . 10 (((𝜑𝑓𝑎) ∧ 𝑑𝑎) → (((𝐹𝐴) ↾ 𝑎)‘𝑓) = ((𝐹𝑎)‘𝑓))
51 fvres 6861 . . . . . . . . . . 11 (𝑓𝑎 → ((𝐹𝑎)‘𝑓) = (𝐹𝑓))
5251ad2antlr 725 . . . . . . . . . 10 (((𝜑𝑓𝑎) ∧ 𝑑𝑎) → ((𝐹𝑎)‘𝑓) = (𝐹𝑓))
5350, 52eqtrd 2776 . . . . . . . . 9 (((𝜑𝑓𝑎) ∧ 𝑑𝑎) → (((𝐹𝐴) ↾ 𝑎)‘𝑓) = (𝐹𝑓))
5449, 53breq12d 5118 . . . . . . . 8 (((𝜑𝑓𝑎) ∧ 𝑑𝑎) → ((((𝐹𝐴) ↾ 𝑎)‘𝑑)𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓) ↔ (𝐹𝑑)𝑅(𝐹𝑓)))
5554notbid 317 . . . . . . 7 (((𝜑𝑓𝑎) ∧ 𝑑𝑎) → (¬ (((𝐹𝐴) ↾ 𝑎)‘𝑑)𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓) ↔ ¬ (𝐹𝑑)𝑅(𝐹𝑓)))
5655ralbidva 3172 . . . . . 6 ((𝜑𝑓𝑎) → (∀𝑑𝑎 ¬ (((𝐹𝐴) ↾ 𝑎)‘𝑑)𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓) ↔ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓)))
5743, 56bitrd 278 . . . . 5 ((𝜑𝑓𝑎) → (∀𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓) ↔ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓)))
5857rexbidva 3173 . . . 4 (𝜑 → (∃𝑓𝑎𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓) ↔ ∃𝑓𝑎𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓)))
5936, 58bitrd 278 . . 3 (𝜑 → (∃𝑑 ∈ ((𝐹𝐴) “ 𝑎)∀𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅𝑑 ↔ ∃𝑓𝑎𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓)))
603inex1 5274 . . . . . . 7 (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ∈ V
6160a1i 11 . . . . . 6 ((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) → (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ∈ V)
6213sselda 3944 . . . . . . . 8 ((𝜑𝑓𝑎) → 𝑓𝐴)
63 fnwe2.su . . . . . . . . . 10 (𝑧 = (𝐹𝑥) → 𝑆 = 𝑈)
64 fnwe2.t . . . . . . . . . 10 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑈𝑦))}
65 fnwe2.s . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})
6663, 64, 65fnwe2lem1 41363 . . . . . . . . 9 ((𝜑𝑓𝐴) → (𝐹𝑓) / 𝑧𝑆 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)})
67 wefr 5623 . . . . . . . . 9 ((𝐹𝑓) / 𝑧𝑆 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)} → (𝐹𝑓) / 𝑧𝑆 Fr {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)})
6866, 67syl 17 . . . . . . . 8 ((𝜑𝑓𝐴) → (𝐹𝑓) / 𝑧𝑆 Fr {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)})
6962, 68syldan 591 . . . . . . 7 ((𝜑𝑓𝑎) → (𝐹𝑓) / 𝑧𝑆 Fr {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)})
7069adantrr 715 . . . . . 6 ((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) → (𝐹𝑓) / 𝑧𝑆 Fr {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)})
71 inss2 4189 . . . . . . 7 (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ⊆ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}
7271a1i 11 . . . . . 6 ((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) → (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ⊆ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)})
73 simprl 769 . . . . . . . 8 ((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) → 𝑓𝑎)
74 fveqeq2 6851 . . . . . . . . 9 (𝑦 = 𝑓 → ((𝐹𝑦) = (𝐹𝑓) ↔ (𝐹𝑓) = (𝐹𝑓)))
7562adantrr 715 . . . . . . . . 9 ((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) → 𝑓𝐴)
76 eqidd 2737 . . . . . . . . 9 ((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) → (𝐹𝑓) = (𝐹𝑓))
7774, 75, 76elrabd 3647 . . . . . . . 8 ((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) → 𝑓 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)})
7873, 77elind 4154 . . . . . . 7 ((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) → 𝑓 ∈ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}))
7978ne0d 4295 . . . . . 6 ((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) → (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ≠ ∅)
80 fri 5593 . . . . . 6 ((((𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ∈ V ∧ (𝐹𝑓) / 𝑧𝑆 Fr {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ∧ ((𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ⊆ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)} ∧ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ≠ ∅)) → ∃𝑒 ∈ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)})∀𝑔 ∈ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)
8161, 70, 72, 79, 80syl22anc 837 . . . . 5 ((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) → ∃𝑒 ∈ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)})∀𝑔 ∈ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)
82 elin 3926 . . . . . . . 8 (𝑒 ∈ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ↔ (𝑒𝑎𝑒 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}))
83 fveqeq2 6851 . . . . . . . . . 10 (𝑦 = 𝑒 → ((𝐹𝑦) = (𝐹𝑓) ↔ (𝐹𝑒) = (𝐹𝑓)))
8483elrab 3645 . . . . . . . . 9 (𝑒 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)} ↔ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))
8584anbi2i 623 . . . . . . . 8 ((𝑒𝑎𝑒 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ↔ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓))))
8682, 85bitri 274 . . . . . . 7 (𝑒 ∈ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ↔ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓))))
87 elin 3926 . . . . . . . . . . . . 13 (𝑔 ∈ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ↔ (𝑔𝑎𝑔 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}))
88 fveqeq2 6851 . . . . . . . . . . . . . . 15 (𝑦 = 𝑔 → ((𝐹𝑦) = (𝐹𝑓) ↔ (𝐹𝑔) = (𝐹𝑓)))
8988elrab 3645 . . . . . . . . . . . . . 14 (𝑔 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)} ↔ (𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)))
9089anbi2i 623 . . . . . . . . . . . . 13 ((𝑔𝑎𝑔 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ↔ (𝑔𝑎 ∧ (𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓))))
9187, 90bitri 274 . . . . . . . . . . . 12 (𝑔 ∈ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ↔ (𝑔𝑎 ∧ (𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓))))
9291imbi1i 349 . . . . . . . . . . 11 ((𝑔 ∈ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒) ↔ ((𝑔𝑎 ∧ (𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓))) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒))
93 impexp 451 . . . . . . . . . . 11 (((𝑔𝑎 ∧ (𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓))) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒) ↔ (𝑔𝑎 → ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)))
9492, 93bitri 274 . . . . . . . . . 10 ((𝑔 ∈ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒) ↔ (𝑔𝑎 → ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)))
9594ralbii2 3092 . . . . . . . . 9 (∀𝑔 ∈ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒 ↔ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒))
96 simplrl 775 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) → 𝑒𝑎)
97 fveq2 6842 . . . . . . . . . . . . . . . . . 18 (𝑑 = 𝑐 → (𝐹𝑑) = (𝐹𝑐))
9897breq1d 5115 . . . . . . . . . . . . . . . . 17 (𝑑 = 𝑐 → ((𝐹𝑑)𝑅(𝐹𝑓) ↔ (𝐹𝑐)𝑅(𝐹𝑓)))
9998notbid 317 . . . . . . . . . . . . . . . 16 (𝑑 = 𝑐 → (¬ (𝐹𝑑)𝑅(𝐹𝑓) ↔ ¬ (𝐹𝑐)𝑅(𝐹𝑓)))
100 simplrr 776 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) → ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))
101100ad2antrr 724 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) ∧ 𝑐𝑎) → ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))
102 simpr 485 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) ∧ 𝑐𝑎) → 𝑐𝑎)
10399, 101, 102rspcdva 3582 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) ∧ 𝑐𝑎) → ¬ (𝐹𝑐)𝑅(𝐹𝑓))
104 simprrr 780 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) → (𝐹𝑒) = (𝐹𝑓))
105104ad2antrr 724 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) ∧ 𝑐𝑎) → (𝐹𝑒) = (𝐹𝑓))
106105breq2d 5117 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) ∧ 𝑐𝑎) → ((𝐹𝑐)𝑅(𝐹𝑒) ↔ (𝐹𝑐)𝑅(𝐹𝑓)))
107103, 106mtbird 324 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) ∧ 𝑐𝑎) → ¬ (𝐹𝑐)𝑅(𝐹𝑒))
10813ad3antrrr 728 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) → 𝑎𝐴)
109108sselda 3944 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) ∧ 𝑐𝑎) → 𝑐𝐴)
110109adantrr 715 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) ∧ (𝑐𝑎 ∧ (𝐹𝑐) = (𝐹𝑒))) → 𝑐𝐴)
111 simprr 771 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) ∧ (𝑐𝑎 ∧ (𝐹𝑐) = (𝐹𝑒))) → (𝐹𝑐) = (𝐹𝑒))
112104ad2antrr 724 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) ∧ (𝑐𝑎 ∧ (𝐹𝑐) = (𝐹𝑒))) → (𝐹𝑒) = (𝐹𝑓))
113111, 112eqtrd 2776 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) ∧ (𝑐𝑎 ∧ (𝐹𝑐) = (𝐹𝑒))) → (𝐹𝑐) = (𝐹𝑓))
114 eleq1w 2820 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = 𝑐 → (𝑔𝐴𝑐𝐴))
115 fveqeq2 6851 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = 𝑐 → ((𝐹𝑔) = (𝐹𝑓) ↔ (𝐹𝑐) = (𝐹𝑓)))
116114, 115anbi12d 631 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = 𝑐 → ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) ↔ (𝑐𝐴 ∧ (𝐹𝑐) = (𝐹𝑓))))
117 breq1 5108 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = 𝑐 → (𝑔(𝐹𝑓) / 𝑧𝑆𝑒𝑐(𝐹𝑓) / 𝑧𝑆𝑒))
118117notbid 317 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = 𝑐 → (¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒 ↔ ¬ 𝑐(𝐹𝑓) / 𝑧𝑆𝑒))
119116, 118imbi12d 344 . . . . . . . . . . . . . . . . . . 19 (𝑔 = 𝑐 → (((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒) ↔ ((𝑐𝐴 ∧ (𝐹𝑐) = (𝐹𝑓)) → ¬ 𝑐(𝐹𝑓) / 𝑧𝑆𝑒)))
120 simplr 767 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) ∧ (𝑐𝑎 ∧ (𝐹𝑐) = (𝐹𝑒))) → ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒))
121 simprl 769 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) ∧ (𝑐𝑎 ∧ (𝐹𝑐) = (𝐹𝑒))) → 𝑐𝑎)
122119, 120, 121rspcdva 3582 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) ∧ (𝑐𝑎 ∧ (𝐹𝑐) = (𝐹𝑒))) → ((𝑐𝐴 ∧ (𝐹𝑐) = (𝐹𝑓)) → ¬ 𝑐(𝐹𝑓) / 𝑧𝑆𝑒))
123110, 113, 122mp2and 697 . . . . . . . . . . . . . . . . 17 (((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) ∧ (𝑐𝑎 ∧ (𝐹𝑐) = (𝐹𝑒))) → ¬ 𝑐(𝐹𝑓) / 𝑧𝑆𝑒)
124111, 112eqtr2d 2777 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) ∧ (𝑐𝑎 ∧ (𝐹𝑐) = (𝐹𝑒))) → (𝐹𝑓) = (𝐹𝑐))
125124csbeq1d 3859 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) ∧ (𝑐𝑎 ∧ (𝐹𝑐) = (𝐹𝑒))) → (𝐹𝑓) / 𝑧𝑆 = (𝐹𝑐) / 𝑧𝑆)
126125breqd 5116 . . . . . . . . . . . . . . . . 17 (((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) ∧ (𝑐𝑎 ∧ (𝐹𝑐) = (𝐹𝑒))) → (𝑐(𝐹𝑓) / 𝑧𝑆𝑒𝑐(𝐹𝑐) / 𝑧𝑆𝑒))
127123, 126mtbid 323 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) ∧ (𝑐𝑎 ∧ (𝐹𝑐) = (𝐹𝑒))) → ¬ 𝑐(𝐹𝑐) / 𝑧𝑆𝑒)
128127expr 457 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) ∧ 𝑐𝑎) → ((𝐹𝑐) = (𝐹𝑒) → ¬ 𝑐(𝐹𝑐) / 𝑧𝑆𝑒))
129 imnan 400 . . . . . . . . . . . . . . 15 (((𝐹𝑐) = (𝐹𝑒) → ¬ 𝑐(𝐹𝑐) / 𝑧𝑆𝑒) ↔ ¬ ((𝐹𝑐) = (𝐹𝑒) ∧ 𝑐(𝐹𝑐) / 𝑧𝑆𝑒))
130128, 129sylib 217 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) ∧ 𝑐𝑎) → ¬ ((𝐹𝑐) = (𝐹𝑒) ∧ 𝑐(𝐹𝑐) / 𝑧𝑆𝑒))
131 ioran 982 . . . . . . . . . . . . . 14 (¬ ((𝐹𝑐)𝑅(𝐹𝑒) ∨ ((𝐹𝑐) = (𝐹𝑒) ∧ 𝑐(𝐹𝑐) / 𝑧𝑆𝑒)) ↔ (¬ (𝐹𝑐)𝑅(𝐹𝑒) ∧ ¬ ((𝐹𝑐) = (𝐹𝑒) ∧ 𝑐(𝐹𝑐) / 𝑧𝑆𝑒)))
132107, 130, 131sylanbrc 583 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) ∧ 𝑐𝑎) → ¬ ((𝐹𝑐)𝑅(𝐹𝑒) ∨ ((𝐹𝑐) = (𝐹𝑒) ∧ 𝑐(𝐹𝑐) / 𝑧𝑆𝑒)))
13363, 64fnwe2val 41362 . . . . . . . . . . . . 13 (𝑐𝑇𝑒 ↔ ((𝐹𝑐)𝑅(𝐹𝑒) ∨ ((𝐹𝑐) = (𝐹𝑒) ∧ 𝑐(𝐹𝑐) / 𝑧𝑆𝑒)))
134132, 133sylnibr 328 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) ∧ 𝑐𝑎) → ¬ 𝑐𝑇𝑒)
135134ralrimiva 3143 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) → ∀𝑐𝑎 ¬ 𝑐𝑇𝑒)
136 breq2 5109 . . . . . . . . . . . . . 14 (𝑏 = 𝑒 → (𝑐𝑇𝑏𝑐𝑇𝑒))
137136notbid 317 . . . . . . . . . . . . 13 (𝑏 = 𝑒 → (¬ 𝑐𝑇𝑏 ↔ ¬ 𝑐𝑇𝑒))
138137ralbidv 3174 . . . . . . . . . . . 12 (𝑏 = 𝑒 → (∀𝑐𝑎 ¬ 𝑐𝑇𝑏 ↔ ∀𝑐𝑎 ¬ 𝑐𝑇𝑒))
139138rspcev 3581 . . . . . . . . . . 11 ((𝑒𝑎 ∧ ∀𝑐𝑎 ¬ 𝑐𝑇𝑒) → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑇𝑏)
14096, 135, 139syl2anc 584 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑇𝑏)
141140ex 413 . . . . . . . . 9 (((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) → (∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒) → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑇𝑏))
14295, 141biimtrid 241 . . . . . . . 8 (((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) → (∀𝑔 ∈ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒 → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑇𝑏))
143142ex 413 . . . . . . 7 ((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) → ((𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓))) → (∀𝑔 ∈ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒 → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑇𝑏)))
14486, 143biimtrid 241 . . . . . 6 ((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) → (𝑒 ∈ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) → (∀𝑔 ∈ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒 → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑇𝑏)))
145144rexlimdv 3150 . . . . 5 ((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) → (∃𝑒 ∈ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)})∀𝑔 ∈ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒 → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑇𝑏))
14681, 145mpd 15 . . . 4 ((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑇𝑏)
147146rexlimdvaa 3153 . . 3 (𝜑 → (∃𝑓𝑎𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓) → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑇𝑏))
14859, 147sylbid 239 . 2 (𝜑 → (∃𝑑 ∈ ((𝐹𝐴) “ 𝑎)∀𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅𝑑 → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑇𝑏))
14925, 148mpd 15 1 (𝜑 → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑇𝑏)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  csb 3855  cin 3909  wss 3910  c0 4282   class class class wbr 5105  {copab 5167   Fr wfr 5585   We wwe 5587  dom cdm 5633  ran crn 5634  cres 5635  cima 5636  Fun wfun 6490   Fn wfn 6491  wf 6492  cfv 6496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504
This theorem is referenced by:  fnwe2  41366
  Copyright terms: Public domain W3C validator