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Theorem fnwe2lem2 42617
Description: Lemma for fnwe2 42619. 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 6726 . . . 4 ((𝐹𝐴):𝐴𝐵 → Fun (𝐹𝐴))
3 vex 3465 . . . . 5 𝑎 ∈ V
43funimaex 6642 . . . 4 (Fun (𝐹𝐴) → ((𝐹𝐴) “ 𝑎) ∈ V)
51, 2, 43syl 18 . . 3 (𝜑 → ((𝐹𝐴) “ 𝑎) ∈ V)
6 fnwe2.r . . . 4 (𝜑𝑅 We 𝐵)
7 wefr 5668 . . . 4 (𝑅 We 𝐵𝑅 Fr 𝐵)
86, 7syl 17 . . 3 (𝜑𝑅 Fr 𝐵)
9 imassrn 6075 . . . 4 ((𝐹𝐴) “ 𝑎) ⊆ ran (𝐹𝐴)
101frnd 6731 . . . 4 (𝜑 → ran (𝐹𝐴) ⊆ 𝐵)
119, 10sstrid 3988 . . 3 (𝜑 → ((𝐹𝐴) “ 𝑎) ⊆ 𝐵)
12 incom 4199 . . . . . 6 (dom (𝐹𝐴) ∩ 𝑎) = (𝑎 ∩ dom (𝐹𝐴))
13 fnwe2lem2.a . . . . . . . 8 (𝜑𝑎𝐴)
141fdmd 6733 . . . . . . . 8 (𝜑 → dom (𝐹𝐴) = 𝐴)
1513, 14sseqtrrd 4018 . . . . . . 7 (𝜑𝑎 ⊆ dom (𝐹𝐴))
16 dfss2 3962 . . . . . . 7 (𝑎 ⊆ dom (𝐹𝐴) ↔ (𝑎 ∩ dom (𝐹𝐴)) = 𝑎)
1715, 16sylib 217 . . . . . 6 (𝜑 → (𝑎 ∩ dom (𝐹𝐴)) = 𝑎)
1812, 17eqtrid 2777 . . . . 5 (𝜑 → (dom (𝐹𝐴) ∩ 𝑎) = 𝑎)
19 fnwe2lem2.n0 . . . . 5 (𝜑𝑎 ≠ ∅)
2018, 19eqnetrd 2997 . . . 4 (𝜑 → (dom (𝐹𝐴) ∩ 𝑎) ≠ ∅)
21 imadisj 6084 . . . . 5 (((𝐹𝐴) “ 𝑎) = ∅ ↔ (dom (𝐹𝐴) ∩ 𝑎) = ∅)
2221necon3bii 2982 . . . 4 (((𝐹𝐴) “ 𝑎) ≠ ∅ ↔ (dom (𝐹𝐴) ∩ 𝑎) ≠ ∅)
2320, 22sylibr 233 . . 3 (𝜑 → ((𝐹𝐴) “ 𝑎) ≠ ∅)
24 fri 5638 . . 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 5691 . . . . . 6 ((𝐹𝐴) “ 𝑎) = ran ((𝐹𝐴) ↾ 𝑎)
2726rexeqi 3313 . . . . 5 (∃𝑑 ∈ ((𝐹𝐴) “ 𝑎)∀𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅𝑑 ↔ ∃𝑑 ∈ ran ((𝐹𝐴) ↾ 𝑎)∀𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅𝑑)
281ffnd 6724 . . . . . . 7 (𝜑 → (𝐹𝐴) Fn 𝐴)
29 fnssres 6679 . . . . . . 7 (((𝐹𝐴) Fn 𝐴𝑎𝐴) → ((𝐹𝐴) ↾ 𝑎) Fn 𝑎)
3028, 13, 29syl2anc 582 . . . . . 6 (𝜑 → ((𝐹𝐴) ↾ 𝑎) Fn 𝑎)
31 breq2 5153 . . . . . . . . 9 (𝑑 = (((𝐹𝐴) ↾ 𝑎)‘𝑓) → (𝑒𝑅𝑑𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓)))
3231notbid 317 . . . . . . . 8 (𝑑 = (((𝐹𝐴) ↾ 𝑎)‘𝑓) → (¬ 𝑒𝑅𝑑 ↔ ¬ 𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓)))
3332ralbidv 3167 . . . . . . 7 (𝑑 = (((𝐹𝐴) ↾ 𝑎)‘𝑓) → (∀𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅𝑑 ↔ ∀𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓)))
3433rexrn 7096 . . . . . 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 3312 . . . . . . . 8 (∀𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓) ↔ ∀𝑒 ∈ ran ((𝐹𝐴) ↾ 𝑎) ¬ 𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓))
38 breq1 5152 . . . . . . . . . . 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 7097 . . . . . . . . 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 479 . . . . . 6 ((𝜑𝑓𝑎) → (∀𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓) ↔ ∀𝑑𝑎 ¬ (((𝐹𝐴) ↾ 𝑎)‘𝑑)𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓)))
4413resabs1d 6013 . . . . . . . . . . . 12 (𝜑 → ((𝐹𝐴) ↾ 𝑎) = (𝐹𝑎))
4544ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝑓𝑎) ∧ 𝑑𝑎) → ((𝐹𝐴) ↾ 𝑎) = (𝐹𝑎))
4645fveq1d 6898 . . . . . . . . . 10 (((𝜑𝑓𝑎) ∧ 𝑑𝑎) → (((𝐹𝐴) ↾ 𝑎)‘𝑑) = ((𝐹𝑎)‘𝑑))
47 fvres 6915 . . . . . . . . . . 11 (𝑑𝑎 → ((𝐹𝑎)‘𝑑) = (𝐹𝑑))
4847adantl 480 . . . . . . . . . 10 (((𝜑𝑓𝑎) ∧ 𝑑𝑎) → ((𝐹𝑎)‘𝑑) = (𝐹𝑑))
4946, 48eqtrd 2765 . . . . . . . . 9 (((𝜑𝑓𝑎) ∧ 𝑑𝑎) → (((𝐹𝐴) ↾ 𝑎)‘𝑑) = (𝐹𝑑))
5045fveq1d 6898 . . . . . . . . . 10 (((𝜑𝑓𝑎) ∧ 𝑑𝑎) → (((𝐹𝐴) ↾ 𝑎)‘𝑓) = ((𝐹𝑎)‘𝑓))
51 fvres 6915 . . . . . . . . . . 11 (𝑓𝑎 → ((𝐹𝑎)‘𝑓) = (𝐹𝑓))
5251ad2antlr 725 . . . . . . . . . 10 (((𝜑𝑓𝑎) ∧ 𝑑𝑎) → ((𝐹𝑎)‘𝑓) = (𝐹𝑓))
5350, 52eqtrd 2765 . . . . . . . . 9 (((𝜑𝑓𝑎) ∧ 𝑑𝑎) → (((𝐹𝐴) ↾ 𝑎)‘𝑓) = (𝐹𝑓))
5449, 53breq12d 5162 . . . . . . . 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 3165 . . . . . 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 3166 . . . 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 5318 . . . . . . 7 (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ∈ V
6160a1i 11 . . . . . 6 ((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) → (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ∈ V)
6213sselda 3976 . . . . . . . 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 42616 . . . . . . . . 9 ((𝜑𝑓𝐴) → (𝐹𝑓) / 𝑧𝑆 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)})
67 wefr 5668 . . . . . . . . 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 589 . . . . . . 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 4228 . . . . . . 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 6905 . . . . . . . . 9 (𝑦 = 𝑓 → ((𝐹𝑦) = (𝐹𝑓) ↔ (𝐹𝑓) = (𝐹𝑓)))
7562adantrr 715 . . . . . . . . 9 ((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) → 𝑓𝐴)
76 eqidd 2726 . . . . . . . . 9 ((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) → (𝐹𝑓) = (𝐹𝑓))
7774, 75, 76elrabd 3681 . . . . . . . 8 ((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) → 𝑓 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)})
7873, 77elind 4192 . . . . . . 7 ((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) → 𝑓 ∈ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}))
7978ne0d 4335 . . . . . 6 ((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) → (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ≠ ∅)
80 fri 5638 . . . . . 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 3960 . . . . . . . 8 (𝑒 ∈ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ↔ (𝑒𝑎𝑒 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}))
83 fveqeq2 6905 . . . . . . . . . 10 (𝑦 = 𝑒 → ((𝐹𝑦) = (𝐹𝑓) ↔ (𝐹𝑒) = (𝐹𝑓)))
8483elrab 3679 . . . . . . . . 9 (𝑒 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)} ↔ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))
8584anbi2i 621 . . . . . . . 8 ((𝑒𝑎𝑒 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ↔ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓))))
8682, 85bitri 274 . . . . . . 7 (𝑒 ∈ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ↔ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓))))
87 elin 3960 . . . . . . . . . . . . 13 (𝑔 ∈ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ↔ (𝑔𝑎𝑔 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}))
88 fveqeq2 6905 . . . . . . . . . . . . . . 15 (𝑦 = 𝑔 → ((𝐹𝑦) = (𝐹𝑓) ↔ (𝐹𝑔) = (𝐹𝑓)))
8988elrab 3679 . . . . . . . . . . . . . 14 (𝑔 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)} ↔ (𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)))
9089anbi2i 621 . . . . . . . . . . . . 13 ((𝑔𝑎𝑔 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ↔ (𝑔𝑎 ∧ (𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓))))
9187, 90bitri 274 . . . . . . . . . . . 12 (𝑔 ∈ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ↔ (𝑔𝑎 ∧ (𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓))))
9291imbi1i 348 . . . . . . . . . . 11 ((𝑔 ∈ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒) ↔ ((𝑔𝑎 ∧ (𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓))) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒))
93 impexp 449 . . . . . . . . . . 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 3078 . . . . . . . . 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 6896 . . . . . . . . . . . . . . . . . 18 (𝑑 = 𝑐 → (𝐹𝑑) = (𝐹𝑐))
9897breq1d 5159 . . . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) ∧ 𝑐𝑎) → 𝑐𝑎)
10399, 101, 102rspcdva 3607 . . . . . . . . . . . . . . 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 5161 . . . . . . . . . . . . . . 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 3976 . . . . . . . . . . . . . . . . . . 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 2765 . . . . . . . . . . . . . . . . . 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 2808 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = 𝑐 → (𝑔𝐴𝑐𝐴))
115 fveqeq2 6905 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = 𝑐 → ((𝐹𝑔) = (𝐹𝑓) ↔ (𝐹𝑐) = (𝐹𝑓)))
116114, 115anbi12d 630 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = 𝑐 → ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) ↔ (𝑐𝐴 ∧ (𝐹𝑐) = (𝐹𝑓))))
117 breq1 5152 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = 𝑐 → (𝑔(𝐹𝑓) / 𝑧𝑆𝑒𝑐(𝐹𝑓) / 𝑧𝑆𝑒))
118117notbid 317 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = 𝑐 → (¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒 ↔ ¬ 𝑐(𝐹𝑓) / 𝑧𝑆𝑒))
119116, 118imbi12d 343 . . . . . . . . . . . . . . . . . . 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 3607 . . . . . . . . . . . . . . . . . 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 2766 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) ∧ (𝑐𝑎 ∧ (𝐹𝑐) = (𝐹𝑒))) → (𝐹𝑓) = (𝐹𝑐))
125124csbeq1d 3893 . . . . . . . . . . . . . . . . . 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 5160 . . . . . . . . . . . . . . . . 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 455 . . . . . . . . . . . . . . 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 398 . . . . . . . . . . . . . . 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 981 . . . . . . . . . . . . . 14 (¬ ((𝐹𝑐)𝑅(𝐹𝑒) ∨ ((𝐹𝑐) = (𝐹𝑒) ∧ 𝑐(𝐹𝑐) / 𝑧𝑆𝑒)) ↔ (¬ (𝐹𝑐)𝑅(𝐹𝑒) ∧ ¬ ((𝐹𝑐) = (𝐹𝑒) ∧ 𝑐(𝐹𝑐) / 𝑧𝑆𝑒)))
132107, 130, 131sylanbrc 581 . . . . . . . . . . . . 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 42615 . . . . . . . . . . . . 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 3135 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) → ∀𝑐𝑎 ¬ 𝑐𝑇𝑒)
136 breq2 5153 . . . . . . . . . . . . . 14 (𝑏 = 𝑒 → (𝑐𝑇𝑏𝑐𝑇𝑒))
137136notbid 317 . . . . . . . . . . . . 13 (𝑏 = 𝑒 → (¬ 𝑐𝑇𝑏 ↔ ¬ 𝑐𝑇𝑒))
138137ralbidv 3167 . . . . . . . . . . . 12 (𝑏 = 𝑒 → (∀𝑐𝑎 ¬ 𝑐𝑇𝑏 ↔ ∀𝑐𝑎 ¬ 𝑐𝑇𝑒))
139138rspcev 3606 . . . . . . . . . . 11 ((𝑒𝑎 ∧ ∀𝑐𝑎 ¬ 𝑐𝑇𝑒) → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑇𝑏)
14096, 135, 139syl2anc 582 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑇𝑏)
141140ex 411 . . . . . . . . 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 411 . . . . . . 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 3142 . . . . 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 3145 . . 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 394  wo 845   = wceq 1533  wcel 2098  wne 2929  wral 3050  wrex 3059  {crab 3418  Vcvv 3461  csb 3889  cin 3943  wss 3944  c0 4322   class class class wbr 5149  {copab 5211   Fr wfr 5630   We wwe 5632  dom cdm 5678  ran crn 5679  cres 5680  cima 5681  Fun wfun 6543   Fn wfn 6544  wf 6545  cfv 6549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557
This theorem is referenced by:  fnwe2  42619
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