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Theorem fnwe2lem2 41188
Description: Lemma for fnwe2 41190. 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 6659 . . . 4 ((𝐹𝐴):𝐴𝐵 → Fun (𝐹𝐴))
3 vex 3446 . . . . 5 𝑎 ∈ V
43funimaex 6577 . . . 4 (Fun (𝐹𝐴) → ((𝐹𝐴) “ 𝑎) ∈ V)
51, 2, 43syl 18 . . 3 (𝜑 → ((𝐹𝐴) “ 𝑎) ∈ V)
6 fnwe2.r . . . 4 (𝜑𝑅 We 𝐵)
7 wefr 5615 . . . 4 (𝑅 We 𝐵𝑅 Fr 𝐵)
86, 7syl 17 . . 3 (𝜑𝑅 Fr 𝐵)
9 imassrn 6015 . . . 4 ((𝐹𝐴) “ 𝑎) ⊆ ran (𝐹𝐴)
101frnd 6664 . . . 4 (𝜑 → ran (𝐹𝐴) ⊆ 𝐵)
119, 10sstrid 3947 . . 3 (𝜑 → ((𝐹𝐴) “ 𝑎) ⊆ 𝐵)
12 incom 4153 . . . . . 6 (dom (𝐹𝐴) ∩ 𝑎) = (𝑎 ∩ dom (𝐹𝐴))
13 fnwe2lem2.a . . . . . . . 8 (𝜑𝑎𝐴)
141fdmd 6667 . . . . . . . 8 (𝜑 → dom (𝐹𝐴) = 𝐴)
1513, 14sseqtrrd 3977 . . . . . . 7 (𝜑𝑎 ⊆ dom (𝐹𝐴))
16 df-ss 3919 . . . . . . 7 (𝑎 ⊆ dom (𝐹𝐴) ↔ (𝑎 ∩ dom (𝐹𝐴)) = 𝑎)
1715, 16sylib 217 . . . . . 6 (𝜑 → (𝑎 ∩ dom (𝐹𝐴)) = 𝑎)
1812, 17eqtrid 2789 . . . . 5 (𝜑 → (dom (𝐹𝐴) ∩ 𝑎) = 𝑎)
19 fnwe2lem2.n0 . . . . 5 (𝜑𝑎 ≠ ∅)
2018, 19eqnetrd 3009 . . . 4 (𝜑 → (dom (𝐹𝐴) ∩ 𝑎) ≠ ∅)
21 imadisj 6023 . . . . 5 (((𝐹𝐴) “ 𝑎) = ∅ ↔ (dom (𝐹𝐴) ∩ 𝑎) = ∅)
2221necon3bii 2994 . . . 4 (((𝐹𝐴) “ 𝑎) ≠ ∅ ↔ (dom (𝐹𝐴) ∩ 𝑎) ≠ ∅)
2320, 22sylibr 233 . . 3 (𝜑 → ((𝐹𝐴) “ 𝑎) ≠ ∅)
24 fri 5585 . . 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 5638 . . . . . 6 ((𝐹𝐴) “ 𝑎) = ran ((𝐹𝐴) ↾ 𝑎)
2726rexeqi 3309 . . . . 5 (∃𝑑 ∈ ((𝐹𝐴) “ 𝑎)∀𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅𝑑 ↔ ∃𝑑 ∈ ran ((𝐹𝐴) ↾ 𝑎)∀𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅𝑑)
281ffnd 6657 . . . . . . 7 (𝜑 → (𝐹𝐴) Fn 𝐴)
29 fnssres 6612 . . . . . . 7 (((𝐹𝐴) Fn 𝐴𝑎𝐴) → ((𝐹𝐴) ↾ 𝑎) Fn 𝑎)
3028, 13, 29syl2anc 585 . . . . . 6 (𝜑 → ((𝐹𝐴) ↾ 𝑎) Fn 𝑎)
31 breq2 5101 . . . . . . . . 9 (𝑑 = (((𝐹𝐴) ↾ 𝑎)‘𝑓) → (𝑒𝑅𝑑𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓)))
3231notbid 318 . . . . . . . 8 (𝑑 = (((𝐹𝐴) ↾ 𝑎)‘𝑓) → (¬ 𝑒𝑅𝑑 ↔ ¬ 𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓)))
3332ralbidv 3171 . . . . . . 7 (𝑑 = (((𝐹𝐴) ↾ 𝑎)‘𝑓) → (∀𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅𝑑 ↔ ∀𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓)))
3433rexrn 7024 . . . . . 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 283 . . . 4 (𝜑 → (∃𝑑 ∈ ((𝐹𝐴) “ 𝑎)∀𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅𝑑 ↔ ∃𝑓𝑎𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓)))
3726raleqi 3308 . . . . . . . 8 (∀𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓) ↔ ∀𝑒 ∈ ran ((𝐹𝐴) ↾ 𝑎) ¬ 𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓))
38 breq1 5100 . . . . . . . . . . 11 (𝑒 = (((𝐹𝐴) ↾ 𝑎)‘𝑑) → (𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓) ↔ (((𝐹𝐴) ↾ 𝑎)‘𝑑)𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓)))
3938notbid 318 . . . . . . . . . 10 (𝑒 = (((𝐹𝐴) ↾ 𝑎)‘𝑑) → (¬ 𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓) ↔ ¬ (((𝐹𝐴) ↾ 𝑎)‘𝑑)𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓)))
4039ralrn 7025 . . . . . . . . 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 283 . . . . . . 7 (𝜑 → (∀𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓) ↔ ∀𝑑𝑎 ¬ (((𝐹𝐴) ↾ 𝑎)‘𝑑)𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓)))
4342adantr 482 . . . . . 6 ((𝜑𝑓𝑎) → (∀𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓) ↔ ∀𝑑𝑎 ¬ (((𝐹𝐴) ↾ 𝑎)‘𝑑)𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓)))
4413resabs1d 5959 . . . . . . . . . . . 12 (𝜑 → ((𝐹𝐴) ↾ 𝑎) = (𝐹𝑎))
4544ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝑓𝑎) ∧ 𝑑𝑎) → ((𝐹𝐴) ↾ 𝑎) = (𝐹𝑎))
4645fveq1d 6832 . . . . . . . . . 10 (((𝜑𝑓𝑎) ∧ 𝑑𝑎) → (((𝐹𝐴) ↾ 𝑎)‘𝑑) = ((𝐹𝑎)‘𝑑))
47 fvres 6849 . . . . . . . . . . 11 (𝑑𝑎 → ((𝐹𝑎)‘𝑑) = (𝐹𝑑))
4847adantl 483 . . . . . . . . . 10 (((𝜑𝑓𝑎) ∧ 𝑑𝑎) → ((𝐹𝑎)‘𝑑) = (𝐹𝑑))
4946, 48eqtrd 2777 . . . . . . . . 9 (((𝜑𝑓𝑎) ∧ 𝑑𝑎) → (((𝐹𝐴) ↾ 𝑎)‘𝑑) = (𝐹𝑑))
5045fveq1d 6832 . . . . . . . . . 10 (((𝜑𝑓𝑎) ∧ 𝑑𝑎) → (((𝐹𝐴) ↾ 𝑎)‘𝑓) = ((𝐹𝑎)‘𝑓))
51 fvres 6849 . . . . . . . . . . 11 (𝑓𝑎 → ((𝐹𝑎)‘𝑓) = (𝐹𝑓))
5251ad2antlr 725 . . . . . . . . . 10 (((𝜑𝑓𝑎) ∧ 𝑑𝑎) → ((𝐹𝑎)‘𝑓) = (𝐹𝑓))
5350, 52eqtrd 2777 . . . . . . . . 9 (((𝜑𝑓𝑎) ∧ 𝑑𝑎) → (((𝐹𝐴) ↾ 𝑎)‘𝑓) = (𝐹𝑓))
5449, 53breq12d 5110 . . . . . . . 8 (((𝜑𝑓𝑎) ∧ 𝑑𝑎) → ((((𝐹𝐴) ↾ 𝑎)‘𝑑)𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓) ↔ (𝐹𝑑)𝑅(𝐹𝑓)))
5554notbid 318 . . . . . . 7 (((𝜑𝑓𝑎) ∧ 𝑑𝑎) → (¬ (((𝐹𝐴) ↾ 𝑎)‘𝑑)𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓) ↔ ¬ (𝐹𝑑)𝑅(𝐹𝑓)))
5655ralbidva 3169 . . . . . 6 ((𝜑𝑓𝑎) → (∀𝑑𝑎 ¬ (((𝐹𝐴) ↾ 𝑎)‘𝑑)𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓) ↔ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓)))
5743, 56bitrd 279 . . . . 5 ((𝜑𝑓𝑎) → (∀𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓) ↔ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓)))
5857rexbidva 3170 . . . 4 (𝜑 → (∃𝑓𝑎𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅(((𝐹𝐴) ↾ 𝑎)‘𝑓) ↔ ∃𝑓𝑎𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓)))
5936, 58bitrd 279 . . 3 (𝜑 → (∃𝑑 ∈ ((𝐹𝐴) “ 𝑎)∀𝑒 ∈ ((𝐹𝐴) “ 𝑎) ¬ 𝑒𝑅𝑑 ↔ ∃𝑓𝑎𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓)))
603inex1 5266 . . . . . . 7 (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ∈ V
6160a1i 11 . . . . . 6 ((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) → (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ∈ V)
6213sselda 3936 . . . . . . . 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 41187 . . . . . . . . 9 ((𝜑𝑓𝐴) → (𝐹𝑓) / 𝑧𝑆 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)})
67 wefr 5615 . . . . . . . . 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 592 . . . . . . 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 4181 . . . . . . 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 6839 . . . . . . . . 9 (𝑦 = 𝑓 → ((𝐹𝑦) = (𝐹𝑓) ↔ (𝐹𝑓) = (𝐹𝑓)))
7562adantrr 715 . . . . . . . . 9 ((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) → 𝑓𝐴)
76 eqidd 2738 . . . . . . . . 9 ((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) → (𝐹𝑓) = (𝐹𝑓))
7774, 75, 76elrabd 3640 . . . . . . . 8 ((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) → 𝑓 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)})
7873, 77elind 4146 . . . . . . 7 ((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) → 𝑓 ∈ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}))
7978ne0d 4287 . . . . . 6 ((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) → (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ≠ ∅)
80 fri 5585 . . . . . 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 3918 . . . . . . . 8 (𝑒 ∈ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ↔ (𝑒𝑎𝑒 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}))
83 fveqeq2 6839 . . . . . . . . . 10 (𝑦 = 𝑒 → ((𝐹𝑦) = (𝐹𝑓) ↔ (𝐹𝑒) = (𝐹𝑓)))
8483elrab 3638 . . . . . . . . 9 (𝑒 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)} ↔ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))
8584anbi2i 624 . . . . . . . 8 ((𝑒𝑎𝑒 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ↔ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓))))
8682, 85bitri 275 . . . . . . 7 (𝑒 ∈ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ↔ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓))))
87 elin 3918 . . . . . . . . . . . . 13 (𝑔 ∈ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ↔ (𝑔𝑎𝑔 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}))
88 fveqeq2 6839 . . . . . . . . . . . . . . 15 (𝑦 = 𝑔 → ((𝐹𝑦) = (𝐹𝑓) ↔ (𝐹𝑔) = (𝐹𝑓)))
8988elrab 3638 . . . . . . . . . . . . . 14 (𝑔 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)} ↔ (𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)))
9089anbi2i 624 . . . . . . . . . . . . 13 ((𝑔𝑎𝑔 ∈ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ↔ (𝑔𝑎 ∧ (𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓))))
9187, 90bitri 275 . . . . . . . . . . . 12 (𝑔 ∈ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) ↔ (𝑔𝑎 ∧ (𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓))))
9291imbi1i 350 . . . . . . . . . . 11 ((𝑔 ∈ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒) ↔ ((𝑔𝑎 ∧ (𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓))) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒))
93 impexp 452 . . . . . . . . . . 11 (((𝑔𝑎 ∧ (𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓))) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒) ↔ (𝑔𝑎 → ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)))
9492, 93bitri 275 . . . . . . . . . 10 ((𝑔 ∈ (𝑎 ∩ {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑓)}) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒) ↔ (𝑔𝑎 → ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)))
9594ralbii2 3089 . . . . . . . . 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 6830 . . . . . . . . . . . . . . . . . 18 (𝑑 = 𝑐 → (𝐹𝑑) = (𝐹𝑐))
9897breq1d 5107 . . . . . . . . . . . . . . . . 17 (𝑑 = 𝑐 → ((𝐹𝑑)𝑅(𝐹𝑓) ↔ (𝐹𝑐)𝑅(𝐹𝑓)))
9998notbid 318 . . . . . . . . . . . . . . . 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 486 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) ∧ 𝑐𝑎) → 𝑐𝑎)
10399, 101, 102rspcdva 3575 . . . . . . . . . . . . . . 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 5109 . . . . . . . . . . . . . . 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 325 . . . . . . . . . . . . . 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 3936 . . . . . . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . . . . . . 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 6839 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = 𝑐 → ((𝐹𝑔) = (𝐹𝑓) ↔ (𝐹𝑐) = (𝐹𝑓)))
116114, 115anbi12d 632 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = 𝑐 → ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) ↔ (𝑐𝐴 ∧ (𝐹𝑐) = (𝐹𝑓))))
117 breq1 5100 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = 𝑐 → (𝑔(𝐹𝑓) / 𝑧𝑆𝑒𝑐(𝐹𝑓) / 𝑧𝑆𝑒))
118117notbid 318 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = 𝑐 → (¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒 ↔ ¬ 𝑐(𝐹𝑓) / 𝑧𝑆𝑒))
119116, 118imbi12d 345 . . . . . . . . . . . . . . . . . . 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 3575 . . . . . . . . . . . . . . . . . 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 2778 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) ∧ (𝑐𝑎 ∧ (𝐹𝑐) = (𝐹𝑒))) → (𝐹𝑓) = (𝐹𝑐))
125124csbeq1d 3851 . . . . . . . . . . . . . . . . . 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 5108 . . . . . . . . . . . . . . . . 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 324 . . . . . . . . . . . . . . . 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 458 . . . . . . . . . . . . . . 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 401 . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . 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 41186 . . . . . . . . . . . . 13 (𝑐𝑇𝑒 ↔ ((𝐹𝑐)𝑅(𝐹𝑒) ∨ ((𝐹𝑐) = (𝐹𝑒) ∧ 𝑐(𝐹𝑐) / 𝑧𝑆𝑒)))
134132, 133sylnibr 329 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) ∧ 𝑐𝑎) → ¬ 𝑐𝑇𝑒)
135134ralrimiva 3140 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) → ∀𝑐𝑎 ¬ 𝑐𝑇𝑒)
136 breq2 5101 . . . . . . . . . . . . . 14 (𝑏 = 𝑒 → (𝑐𝑇𝑏𝑐𝑇𝑒))
137136notbid 318 . . . . . . . . . . . . 13 (𝑏 = 𝑒 → (¬ 𝑐𝑇𝑏 ↔ ¬ 𝑐𝑇𝑒))
138137ralbidv 3171 . . . . . . . . . . . 12 (𝑏 = 𝑒 → (∀𝑐𝑎 ¬ 𝑐𝑇𝑏 ↔ ∀𝑐𝑎 ¬ 𝑐𝑇𝑒))
139138rspcev 3574 . . . . . . . . . . 11 ((𝑒𝑎 ∧ ∀𝑐𝑎 ¬ 𝑐𝑇𝑒) → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑇𝑏)
14096, 135, 139syl2anc 585 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓𝑎 ∧ ∀𝑑𝑎 ¬ (𝐹𝑑)𝑅(𝐹𝑓))) ∧ (𝑒𝑎 ∧ (𝑒𝐴 ∧ (𝐹𝑒) = (𝐹𝑓)))) ∧ ∀𝑔𝑎 ((𝑔𝐴 ∧ (𝐹𝑔) = (𝐹𝑓)) → ¬ 𝑔(𝐹𝑓) / 𝑧𝑆𝑒)) → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑇𝑏)
141140ex 414 . . . . . . . . 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 414 . . . . . . 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 3147 . . . . 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 3150 . . 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 397  wo 845   = wceq 1541  wcel 2106  wne 2941  wral 3062  wrex 3071  {crab 3404  Vcvv 3442  csb 3847  cin 3901  wss 3902  c0 4274   class class class wbr 5097  {copab 5159   Fr wfr 5577   We wwe 5579  dom cdm 5625  ran crn 5626  cres 5627  cima 5628  Fun wfun 6478   Fn wfn 6479  wf 6480  cfv 6484
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 2708  ax-rep 5234  ax-sep 5248  ax-nul 5255  ax-pr 5377
This theorem depends on definitions:  df-bi 206  df-an 398  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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3444  df-sbc 3732  df-csb 3848  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4275  df-if 4479  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4858  df-br 5098  df-opab 5160  df-mpt 5181  df-id 5523  df-po 5537  df-so 5538  df-fr 5580  df-we 5582  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6436  df-fun 6486  df-fn 6487  df-f 6488  df-fv 6492
This theorem is referenced by:  fnwe2  41190
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