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Theorem wessf1ornlem 43393
Description: Given a function 𝐹 on a well-ordered domain 𝐴 there exists a subset of 𝐴 such that 𝐹 restricted to such subset is injective and onto the range of 𝐹 (without using the axiom of choice). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
wessf1ornlem.f (𝜑𝐹 Fn 𝐴)
wessf1ornlem.a (𝜑𝐴𝑉)
wessf1ornlem.r (𝜑𝑅 We 𝐴)
wessf1ornlem.g 𝐺 = (𝑦 ∈ ran 𝐹 ↦ (𝑥 ∈ (𝐹 “ {𝑦})∀𝑧 ∈ (𝐹 “ {𝑦}) ¬ 𝑧𝑅𝑥))
Assertion
Ref Expression
wessf1ornlem (𝜑 → ∃𝑥 ∈ 𝒫 𝐴(𝐹𝑥):𝑥1-1-onto→ran 𝐹)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑦,𝑧)   𝐺(𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem wessf1ornlem
Dummy variables 𝑡 𝑢 𝑣 𝑤 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wessf1ornlem.a . . . 4 (𝜑𝐴𝑉)
2 cnvimass 6033 . . . . . . . 8 (𝐹 “ {𝑢}) ⊆ dom 𝐹
3 wessf1ornlem.f . . . . . . . . . 10 (𝜑𝐹 Fn 𝐴)
43fndmd 6607 . . . . . . . . 9 (𝜑 → dom 𝐹 = 𝐴)
54adantr 481 . . . . . . . 8 ((𝜑𝑢 ∈ ran 𝐹) → dom 𝐹 = 𝐴)
62, 5sseqtrid 3996 . . . . . . 7 ((𝜑𝑢 ∈ ran 𝐹) → (𝐹 “ {𝑢}) ⊆ 𝐴)
7 wessf1ornlem.r . . . . . . . . . 10 (𝜑𝑅 We 𝐴)
87adantr 481 . . . . . . . . 9 ((𝜑𝑢 ∈ ran 𝐹) → 𝑅 We 𝐴)
92, 4sseqtrid 3996 . . . . . . . . . . 11 (𝜑 → (𝐹 “ {𝑢}) ⊆ 𝐴)
101, 9ssexd 5281 . . . . . . . . . 10 (𝜑 → (𝐹 “ {𝑢}) ∈ V)
1110adantr 481 . . . . . . . . 9 ((𝜑𝑢 ∈ ran 𝐹) → (𝐹 “ {𝑢}) ∈ V)
12 inisegn0 6050 . . . . . . . . . . 11 (𝑢 ∈ ran 𝐹 ↔ (𝐹 “ {𝑢}) ≠ ∅)
1312biimpi 215 . . . . . . . . . 10 (𝑢 ∈ ran 𝐹 → (𝐹 “ {𝑢}) ≠ ∅)
1413adantl 482 . . . . . . . . 9 ((𝜑𝑢 ∈ ran 𝐹) → (𝐹 “ {𝑢}) ≠ ∅)
15 wereu 5629 . . . . . . . . 9 ((𝑅 We 𝐴 ∧ ((𝐹 “ {𝑢}) ∈ V ∧ (𝐹 “ {𝑢}) ⊆ 𝐴 ∧ (𝐹 “ {𝑢}) ≠ ∅)) → ∃!𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)
168, 11, 6, 14, 15syl13anc 1372 . . . . . . . 8 ((𝜑𝑢 ∈ ran 𝐹) → ∃!𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)
17 riotacl 7331 . . . . . . . 8 (∃!𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣 → (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) ∈ (𝐹 “ {𝑢}))
1816, 17syl 17 . . . . . . 7 ((𝜑𝑢 ∈ ran 𝐹) → (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) ∈ (𝐹 “ {𝑢}))
196, 18sseldd 3945 . . . . . 6 ((𝜑𝑢 ∈ ran 𝐹) → (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) ∈ 𝐴)
2019ralrimiva 3143 . . . . 5 (𝜑 → ∀𝑢 ∈ ran 𝐹(𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) ∈ 𝐴)
21 wessf1ornlem.g . . . . . . 7 𝐺 = (𝑦 ∈ ran 𝐹 ↦ (𝑥 ∈ (𝐹 “ {𝑦})∀𝑧 ∈ (𝐹 “ {𝑦}) ¬ 𝑧𝑅𝑥))
22 sneq 4596 . . . . . . . . . . 11 (𝑦 = 𝑢 → {𝑦} = {𝑢})
2322imaeq2d 6013 . . . . . . . . . 10 (𝑦 = 𝑢 → (𝐹 “ {𝑦}) = (𝐹 “ {𝑢}))
2423raleqdv 3313 . . . . . . . . . 10 (𝑦 = 𝑢 → (∀𝑧 ∈ (𝐹 “ {𝑦}) ¬ 𝑧𝑅𝑥 ↔ ∀𝑧 ∈ (𝐹 “ {𝑢}) ¬ 𝑧𝑅𝑥))
2523, 24riotaeqbidv 7316 . . . . . . . . 9 (𝑦 = 𝑢 → (𝑥 ∈ (𝐹 “ {𝑦})∀𝑧 ∈ (𝐹 “ {𝑦}) ¬ 𝑧𝑅𝑥) = (𝑥 ∈ (𝐹 “ {𝑢})∀𝑧 ∈ (𝐹 “ {𝑢}) ¬ 𝑧𝑅𝑥))
26 breq1 5108 . . . . . . . . . . . . 13 (𝑧 = 𝑡 → (𝑧𝑅𝑥𝑡𝑅𝑥))
2726notbid 317 . . . . . . . . . . . 12 (𝑧 = 𝑡 → (¬ 𝑧𝑅𝑥 ↔ ¬ 𝑡𝑅𝑥))
2827cbvralvw 3225 . . . . . . . . . . 11 (∀𝑧 ∈ (𝐹 “ {𝑢}) ¬ 𝑧𝑅𝑥 ↔ ∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑥)
29 breq2 5109 . . . . . . . . . . . . 13 (𝑥 = 𝑣 → (𝑡𝑅𝑥𝑡𝑅𝑣))
3029notbid 317 . . . . . . . . . . . 12 (𝑥 = 𝑣 → (¬ 𝑡𝑅𝑥 ↔ ¬ 𝑡𝑅𝑣))
3130ralbidv 3174 . . . . . . . . . . 11 (𝑥 = 𝑣 → (∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑥 ↔ ∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣))
3228, 31bitrid 282 . . . . . . . . . 10 (𝑥 = 𝑣 → (∀𝑧 ∈ (𝐹 “ {𝑢}) ¬ 𝑧𝑅𝑥 ↔ ∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣))
3332cbvriotavw 7323 . . . . . . . . 9 (𝑥 ∈ (𝐹 “ {𝑢})∀𝑧 ∈ (𝐹 “ {𝑢}) ¬ 𝑧𝑅𝑥) = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)
3425, 33eqtrdi 2792 . . . . . . . 8 (𝑦 = 𝑢 → (𝑥 ∈ (𝐹 “ {𝑦})∀𝑧 ∈ (𝐹 “ {𝑦}) ¬ 𝑧𝑅𝑥) = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣))
3534cbvmptv 5218 . . . . . . 7 (𝑦 ∈ ran 𝐹 ↦ (𝑥 ∈ (𝐹 “ {𝑦})∀𝑧 ∈ (𝐹 “ {𝑦}) ¬ 𝑧𝑅𝑥)) = (𝑢 ∈ ran 𝐹 ↦ (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣))
3621, 35eqtri 2764 . . . . . 6 𝐺 = (𝑢 ∈ ran 𝐹 ↦ (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣))
3736rnmptss 7070 . . . . 5 (∀𝑢 ∈ ran 𝐹(𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) ∈ 𝐴 → ran 𝐺𝐴)
3820, 37syl 17 . . . 4 (𝜑 → ran 𝐺𝐴)
391, 38sselpwd 5283 . . 3 (𝜑 → ran 𝐺 ∈ 𝒫 𝐴)
40 dffn3 6681 . . . . . . 7 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
413, 40sylib 217 . . . . . 6 (𝜑𝐹:𝐴⟶ran 𝐹)
4241, 38fssresd 6709 . . . . 5 (𝜑 → (𝐹 ↾ ran 𝐺):ran 𝐺⟶ran 𝐹)
43 fvres 6861 . . . . . . . . . . . . 13 (𝑤 ∈ ran 𝐺 → ((𝐹 ↾ ran 𝐺)‘𝑤) = (𝐹𝑤))
4443eqcomd 2742 . . . . . . . . . . . 12 (𝑤 ∈ ran 𝐺 → (𝐹𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑤))
4544ad2antrr 724 . . . . . . . . . . 11 (((𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ ((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑡)) → (𝐹𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑤))
46 simpr 485 . . . . . . . . . . 11 (((𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ ((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑡)) → ((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑡))
47 fvres 6861 . . . . . . . . . . . 12 (𝑡 ∈ ran 𝐺 → ((𝐹 ↾ ran 𝐺)‘𝑡) = (𝐹𝑡))
4847ad2antlr 725 . . . . . . . . . . 11 (((𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ ((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑡)) → ((𝐹 ↾ ran 𝐺)‘𝑡) = (𝐹𝑡))
4945, 46, 483eqtrd 2780 . . . . . . . . . 10 (((𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ ((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑡)) → (𝐹𝑤) = (𝐹𝑡))
50493adantl1 1166 . . . . . . . . 9 (((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ ((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑡)) → (𝐹𝑤) = (𝐹𝑡))
51 simpl1 1191 . . . . . . . . . . 11 (((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ (𝐹𝑤) = (𝐹𝑡)) → 𝜑)
52 simpl3 1193 . . . . . . . . . . 11 (((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ (𝐹𝑤) = (𝐹𝑡)) → 𝑡 ∈ ran 𝐺)
53 simpl2 1192 . . . . . . . . . . 11 (((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ (𝐹𝑤) = (𝐹𝑡)) → 𝑤 ∈ ran 𝐺)
54 id 22 . . . . . . . . . . . . 13 ((𝐹𝑤) = (𝐹𝑡) → (𝐹𝑤) = (𝐹𝑡))
5554eqcomd 2742 . . . . . . . . . . . 12 ((𝐹𝑤) = (𝐹𝑡) → (𝐹𝑡) = (𝐹𝑤))
5655adantl 482 . . . . . . . . . . 11 (((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ (𝐹𝑤) = (𝐹𝑡)) → (𝐹𝑡) = (𝐹𝑤))
57 eleq1w 2820 . . . . . . . . . . . . . . 15 (𝑏 = 𝑤 → (𝑏 ∈ ran 𝐺𝑤 ∈ ran 𝐺))
58573anbi3d 1442 . . . . . . . . . . . . . 14 (𝑏 = 𝑤 → ((𝜑𝑡 ∈ ran 𝐺𝑏 ∈ ran 𝐺) ↔ (𝜑𝑡 ∈ ran 𝐺𝑤 ∈ ran 𝐺)))
59 fveq2 6842 . . . . . . . . . . . . . . 15 (𝑏 = 𝑤 → (𝐹𝑏) = (𝐹𝑤))
6059eqeq2d 2747 . . . . . . . . . . . . . 14 (𝑏 = 𝑤 → ((𝐹𝑡) = (𝐹𝑏) ↔ (𝐹𝑡) = (𝐹𝑤)))
6158, 60anbi12d 631 . . . . . . . . . . . . 13 (𝑏 = 𝑤 → (((𝜑𝑡 ∈ ran 𝐺𝑏 ∈ ran 𝐺) ∧ (𝐹𝑡) = (𝐹𝑏)) ↔ ((𝜑𝑡 ∈ ran 𝐺𝑤 ∈ ran 𝐺) ∧ (𝐹𝑡) = (𝐹𝑤))))
62 breq1 5108 . . . . . . . . . . . . . 14 (𝑏 = 𝑤 → (𝑏𝑅𝑡𝑤𝑅𝑡))
6362notbid 317 . . . . . . . . . . . . 13 (𝑏 = 𝑤 → (¬ 𝑏𝑅𝑡 ↔ ¬ 𝑤𝑅𝑡))
6461, 63imbi12d 344 . . . . . . . . . . . 12 (𝑏 = 𝑤 → ((((𝜑𝑡 ∈ ran 𝐺𝑏 ∈ ran 𝐺) ∧ (𝐹𝑡) = (𝐹𝑏)) → ¬ 𝑏𝑅𝑡) ↔ (((𝜑𝑡 ∈ ran 𝐺𝑤 ∈ ran 𝐺) ∧ (𝐹𝑡) = (𝐹𝑤)) → ¬ 𝑤𝑅𝑡)))
65 eleq1w 2820 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑡 → (𝑎 ∈ ran 𝐺𝑡 ∈ ran 𝐺))
66653anbi2d 1441 . . . . . . . . . . . . . . 15 (𝑎 = 𝑡 → ((𝜑𝑎 ∈ ran 𝐺𝑏 ∈ ran 𝐺) ↔ (𝜑𝑡 ∈ ran 𝐺𝑏 ∈ ran 𝐺)))
67 fveqeq2 6851 . . . . . . . . . . . . . . 15 (𝑎 = 𝑡 → ((𝐹𝑎) = (𝐹𝑏) ↔ (𝐹𝑡) = (𝐹𝑏)))
6866, 67anbi12d 631 . . . . . . . . . . . . . 14 (𝑎 = 𝑡 → (((𝜑𝑎 ∈ ran 𝐺𝑏 ∈ ran 𝐺) ∧ (𝐹𝑎) = (𝐹𝑏)) ↔ ((𝜑𝑡 ∈ ran 𝐺𝑏 ∈ ran 𝐺) ∧ (𝐹𝑡) = (𝐹𝑏))))
69 breq2 5109 . . . . . . . . . . . . . . 15 (𝑎 = 𝑡 → (𝑏𝑅𝑎𝑏𝑅𝑡))
7069notbid 317 . . . . . . . . . . . . . 14 (𝑎 = 𝑡 → (¬ 𝑏𝑅𝑎 ↔ ¬ 𝑏𝑅𝑡))
7168, 70imbi12d 344 . . . . . . . . . . . . 13 (𝑎 = 𝑡 → ((((𝜑𝑎 ∈ ran 𝐺𝑏 ∈ ran 𝐺) ∧ (𝐹𝑎) = (𝐹𝑏)) → ¬ 𝑏𝑅𝑎) ↔ (((𝜑𝑡 ∈ ran 𝐺𝑏 ∈ ran 𝐺) ∧ (𝐹𝑡) = (𝐹𝑏)) → ¬ 𝑏𝑅𝑡)))
72 eleq1w 2820 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑏 → (𝑡 ∈ ran 𝐺𝑏 ∈ ran 𝐺))
73723anbi3d 1442 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑏 → ((𝜑𝑎 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ↔ (𝜑𝑎 ∈ ran 𝐺𝑏 ∈ ran 𝐺)))
74 fveq2 6842 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑏 → (𝐹𝑡) = (𝐹𝑏))
7574eqeq2d 2747 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑏 → ((𝐹𝑎) = (𝐹𝑡) ↔ (𝐹𝑎) = (𝐹𝑏)))
7673, 75anbi12d 631 . . . . . . . . . . . . . . 15 (𝑡 = 𝑏 → (((𝜑𝑎 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ (𝐹𝑎) = (𝐹𝑡)) ↔ ((𝜑𝑎 ∈ ran 𝐺𝑏 ∈ ran 𝐺) ∧ (𝐹𝑎) = (𝐹𝑏))))
77 breq1 5108 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑏 → (𝑡𝑅𝑎𝑏𝑅𝑎))
7877notbid 317 . . . . . . . . . . . . . . 15 (𝑡 = 𝑏 → (¬ 𝑡𝑅𝑎 ↔ ¬ 𝑏𝑅𝑎))
7976, 78imbi12d 344 . . . . . . . . . . . . . 14 (𝑡 = 𝑏 → ((((𝜑𝑎 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ (𝐹𝑎) = (𝐹𝑡)) → ¬ 𝑡𝑅𝑎) ↔ (((𝜑𝑎 ∈ ran 𝐺𝑏 ∈ ran 𝐺) ∧ (𝐹𝑎) = (𝐹𝑏)) → ¬ 𝑏𝑅𝑎)))
80 eleq1w 2820 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑎 → (𝑤 ∈ ran 𝐺𝑎 ∈ ran 𝐺))
81803anbi2d 1441 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑎 → ((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ↔ (𝜑𝑎 ∈ ran 𝐺𝑡 ∈ ran 𝐺)))
82 fveqeq2 6851 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑎 → ((𝐹𝑤) = (𝐹𝑡) ↔ (𝐹𝑎) = (𝐹𝑡)))
8381, 82anbi12d 631 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑎 → (((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ (𝐹𝑤) = (𝐹𝑡)) ↔ ((𝜑𝑎 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ (𝐹𝑎) = (𝐹𝑡))))
84 breq2 5109 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑎 → (𝑡𝑅𝑤𝑡𝑅𝑎))
8584notbid 317 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑎 → (¬ 𝑡𝑅𝑤 ↔ ¬ 𝑡𝑅𝑎))
8683, 85imbi12d 344 . . . . . . . . . . . . . . 15 (𝑤 = 𝑎 → ((((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ (𝐹𝑤) = (𝐹𝑡)) → ¬ 𝑡𝑅𝑤) ↔ (((𝜑𝑎 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ (𝐹𝑎) = (𝐹𝑡)) → ¬ 𝑡𝑅𝑎)))
8736elrnmpt 5911 . . . . . . . . . . . . . . . . . . . 20 (𝑤 ∈ V → (𝑤 ∈ ran 𝐺 ↔ ∃𝑢 ∈ ran 𝐹 𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)))
8887elv 3451 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ ran 𝐺 ↔ ∃𝑢 ∈ ran 𝐹 𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣))
8988biimpi 215 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ ran 𝐺 → ∃𝑢 ∈ ran 𝐹 𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣))
9089adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ran 𝐺 ∧ (𝐹𝑤) = (𝐹𝑡)) → ∃𝑢 ∈ ran 𝐹 𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣))
91903ad2antl2 1186 . . . . . . . . . . . . . . . 16 (((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ (𝐹𝑤) = (𝐹𝑡)) → ∃𝑢 ∈ ran 𝐹 𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣))
92 simp3 1138 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ 𝑢 ∈ ran 𝐹𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → 𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣))
9392eqcomd 2742 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ 𝑢 ∈ ran 𝐹𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) = 𝑤)
94 simp11 1203 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ 𝑢 ∈ ran 𝐹𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → 𝜑)
95 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) → 𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣))
96 breq2 5109 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑣 = 𝑤 → (𝑡𝑅𝑣𝑡𝑅𝑤))
9796notbid 317 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑣 = 𝑤 → (¬ 𝑡𝑅𝑣 ↔ ¬ 𝑡𝑅𝑤))
9897ralbidv 3174 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑣 = 𝑤 → (∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣 ↔ ∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤))
9998cbvriotavw 7323 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) = (𝑤 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤)
10095, 99eqtr2di 2793 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) → (𝑤 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤) = 𝑤)
1011003ad2ant3 1135 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑢 ∈ ran 𝐹𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → (𝑤 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤) = 𝑤)
10298cbvreuvw 3377 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (∃!𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣 ↔ ∃!𝑤 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤)
10316, 102sylib 217 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑢 ∈ ran 𝐹) → ∃!𝑤 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤)
104 riota1 7335 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (∃!𝑤 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤 → ((𝑤 ∈ (𝐹 “ {𝑢}) ∧ ∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤) ↔ (𝑤 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤) = 𝑤))
105103, 104syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑢 ∈ ran 𝐹) → ((𝑤 ∈ (𝐹 “ {𝑢}) ∧ ∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤) ↔ (𝑤 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤) = 𝑤))
1061053adant3 1132 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑢 ∈ ran 𝐹𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → ((𝑤 ∈ (𝐹 “ {𝑢}) ∧ ∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤) ↔ (𝑤 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤) = 𝑤))
107101, 106mpbird 256 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑢 ∈ ran 𝐹𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → (𝑤 ∈ (𝐹 “ {𝑢}) ∧ ∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤))
108107simpld 495 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑢 ∈ ran 𝐹𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → 𝑤 ∈ (𝐹 “ {𝑢}))
10994, 108syld3an1 1410 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ 𝑢 ∈ ran 𝐹𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → 𝑤 ∈ (𝐹 “ {𝑢}))
110 simp2 1137 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ 𝑢 ∈ ran 𝐹𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → 𝑢 ∈ ran 𝐹)
11194, 110, 16syl2anc 584 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ 𝑢 ∈ ran 𝐹𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → ∃!𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)
11298riota2 7339 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤 ∈ (𝐹 “ {𝑢}) ∧ ∃!𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) → (∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤 ↔ (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) = 𝑤))
113109, 111, 112syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ 𝑢 ∈ ran 𝐹𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → (∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤 ↔ (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) = 𝑤))
11493, 113mpbird 256 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ 𝑢 ∈ ran 𝐹𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → ∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤)
1151143adant1r 1177 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ (𝐹𝑤) = (𝐹𝑡)) ∧ 𝑢 ∈ ran 𝐹𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → ∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤)
11638sselda 3944 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑡 ∈ ran 𝐺) → 𝑡𝐴)
1171163adant2 1131 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) → 𝑡𝐴)
118117adantr 481 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ (𝐹𝑤) = (𝐹𝑡)) → 𝑡𝐴)
1191183ad2ant1 1133 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ (𝐹𝑤) = (𝐹𝑡)) ∧ 𝑢 ∈ ran 𝐹𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → 𝑡𝐴)
12055ad2antlr 725 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ (𝐹𝑤) = (𝐹𝑡)) ∧ 𝑢 ∈ ran 𝐹) → (𝐹𝑡) = (𝐹𝑤))
1211203adant3 1132 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ (𝐹𝑤) = (𝐹𝑡)) ∧ 𝑢 ∈ ran 𝐹𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → (𝐹𝑡) = (𝐹𝑤))
122 fniniseg 7010 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐹 Fn 𝐴 → (𝑤 ∈ (𝐹 “ {𝑢}) ↔ (𝑤𝐴 ∧ (𝐹𝑤) = 𝑢)))
12394, 3, 1223syl 18 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ 𝑢 ∈ ran 𝐹𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → (𝑤 ∈ (𝐹 “ {𝑢}) ↔ (𝑤𝐴 ∧ (𝐹𝑤) = 𝑢)))
124109, 123mpbid 231 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ 𝑢 ∈ ran 𝐹𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → (𝑤𝐴 ∧ (𝐹𝑤) = 𝑢))
125124simprd 496 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ 𝑢 ∈ ran 𝐹𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → (𝐹𝑤) = 𝑢)
1261253adant1r 1177 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ (𝐹𝑤) = (𝐹𝑡)) ∧ 𝑢 ∈ ran 𝐹𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → (𝐹𝑤) = 𝑢)
127121, 126eqtrd 2776 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ (𝐹𝑤) = (𝐹𝑡)) ∧ 𝑢 ∈ ran 𝐹𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → (𝐹𝑡) = 𝑢)
128 fniniseg 7010 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹 Fn 𝐴 → (𝑡 ∈ (𝐹 “ {𝑢}) ↔ (𝑡𝐴 ∧ (𝐹𝑡) = 𝑢)))
1293, 128syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑡 ∈ (𝐹 “ {𝑢}) ↔ (𝑡𝐴 ∧ (𝐹𝑡) = 𝑢)))
1301293ad2ant1 1133 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) → (𝑡 ∈ (𝐹 “ {𝑢}) ↔ (𝑡𝐴 ∧ (𝐹𝑡) = 𝑢)))
131130ad2antrr 724 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ (𝐹𝑤) = (𝐹𝑡)) ∧ 𝑢 ∈ ran 𝐹) → (𝑡 ∈ (𝐹 “ {𝑢}) ↔ (𝑡𝐴 ∧ (𝐹𝑡) = 𝑢)))
1321313adant3 1132 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ (𝐹𝑤) = (𝐹𝑡)) ∧ 𝑢 ∈ ran 𝐹𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → (𝑡 ∈ (𝐹 “ {𝑢}) ↔ (𝑡𝐴 ∧ (𝐹𝑡) = 𝑢)))
133119, 127, 132mpbir2and 711 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ (𝐹𝑤) = (𝐹𝑡)) ∧ 𝑢 ∈ ran 𝐹𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → 𝑡 ∈ (𝐹 “ {𝑢}))
134 rspa 3231 . . . . . . . . . . . . . . . . . 18 ((∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑤𝑡 ∈ (𝐹 “ {𝑢})) → ¬ 𝑡𝑅𝑤)
135115, 133, 134syl2anc 584 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ (𝐹𝑤) = (𝐹𝑡)) ∧ 𝑢 ∈ ran 𝐹𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣)) → ¬ 𝑡𝑅𝑤)
136135rexlimdv3a 3156 . . . . . . . . . . . . . . . 16 (((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ (𝐹𝑤) = (𝐹𝑡)) → (∃𝑢 ∈ ran 𝐹 𝑤 = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) → ¬ 𝑡𝑅𝑤))
13791, 136mpd 15 . . . . . . . . . . . . . . 15 (((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ (𝐹𝑤) = (𝐹𝑡)) → ¬ 𝑡𝑅𝑤)
13886, 137chvarvv 2002 . . . . . . . . . . . . . 14 (((𝜑𝑎 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ (𝐹𝑎) = (𝐹𝑡)) → ¬ 𝑡𝑅𝑎)
13979, 138chvarvv 2002 . . . . . . . . . . . . 13 (((𝜑𝑎 ∈ ran 𝐺𝑏 ∈ ran 𝐺) ∧ (𝐹𝑎) = (𝐹𝑏)) → ¬ 𝑏𝑅𝑎)
14071, 139chvarvv 2002 . . . . . . . . . . . 12 (((𝜑𝑡 ∈ ran 𝐺𝑏 ∈ ran 𝐺) ∧ (𝐹𝑡) = (𝐹𝑏)) → ¬ 𝑏𝑅𝑡)
14164, 140chvarvv 2002 . . . . . . . . . . 11 (((𝜑𝑡 ∈ ran 𝐺𝑤 ∈ ran 𝐺) ∧ (𝐹𝑡) = (𝐹𝑤)) → ¬ 𝑤𝑅𝑡)
14251, 52, 53, 56, 141syl31anc 1373 . . . . . . . . . 10 (((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ (𝐹𝑤) = (𝐹𝑡)) → ¬ 𝑤𝑅𝑡)
143 weso 5624 . . . . . . . . . . . . . 14 (𝑅 We 𝐴𝑅 Or 𝐴)
1447, 143syl 17 . . . . . . . . . . . . 13 (𝜑𝑅 Or 𝐴)
145144adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐹𝑤) = (𝐹𝑡)) → 𝑅 Or 𝐴)
1461453ad2antl1 1185 . . . . . . . . . . 11 (((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ (𝐹𝑤) = (𝐹𝑡)) → 𝑅 Or 𝐴)
14738sselda 3944 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ ran 𝐺) → 𝑤𝐴)
1481473adant3 1132 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) → 𝑤𝐴)
149148adantr 481 . . . . . . . . . . 11 (((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ (𝐹𝑤) = (𝐹𝑡)) → 𝑤𝐴)
150 sotrieq2 5575 . . . . . . . . . . 11 ((𝑅 Or 𝐴 ∧ (𝑤𝐴𝑡𝐴)) → (𝑤 = 𝑡 ↔ (¬ 𝑤𝑅𝑡 ∧ ¬ 𝑡𝑅𝑤)))
151146, 149, 118, 150syl12anc 835 . . . . . . . . . 10 (((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ (𝐹𝑤) = (𝐹𝑡)) → (𝑤 = 𝑡 ↔ (¬ 𝑤𝑅𝑡 ∧ ¬ 𝑡𝑅𝑤)))
152142, 137, 151mpbir2and 711 . . . . . . . . 9 (((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ (𝐹𝑤) = (𝐹𝑡)) → 𝑤 = 𝑡)
15350, 152syldan 591 . . . . . . . 8 (((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) ∧ ((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑡)) → 𝑤 = 𝑡)
154153ex 413 . . . . . . 7 ((𝜑𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺) → (((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑡) → 𝑤 = 𝑡))
1551543expb 1120 . . . . . 6 ((𝜑 ∧ (𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺)) → (((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑡) → 𝑤 = 𝑡))
156155ralrimivva 3197 . . . . 5 (𝜑 → ∀𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺(((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑡) → 𝑤 = 𝑡))
157 dff13 7202 . . . . 5 ((𝐹 ↾ ran 𝐺):ran 𝐺1-1→ran 𝐹 ↔ ((𝐹 ↾ ran 𝐺):ran 𝐺⟶ran 𝐹 ∧ ∀𝑤 ∈ ran 𝐺𝑡 ∈ ran 𝐺(((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘𝑡) → 𝑤 = 𝑡)))
15842, 156, 157sylanbrc 583 . . . 4 (𝜑 → (𝐹 ↾ ran 𝐺):ran 𝐺1-1→ran 𝐹)
159 riotaex 7317 . . . . . . . . . . 11 (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) ∈ V
160159rgenw 3068 . . . . . . . . . 10 𝑢 ∈ ran 𝐹(𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) ∈ V
16136fnmpt 6641 . . . . . . . . . 10 (∀𝑢 ∈ ran 𝐹(𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) ∈ V → 𝐺 Fn ran 𝐹)
162160, 161mp1i 13 . . . . . . . . 9 (𝜑𝐺 Fn ran 𝐹)
163 dffn3 6681 . . . . . . . . 9 (𝐺 Fn ran 𝐹𝐺:ran 𝐹⟶ran 𝐺)
164162, 163sylib 217 . . . . . . . 8 (𝜑𝐺:ran 𝐹⟶ran 𝐺)
165164ffvelcdmda 7035 . . . . . . 7 ((𝜑𝑢 ∈ ran 𝐹) → (𝐺𝑢) ∈ ran 𝐺)
166165fvresd 6862 . . . . . . . 8 ((𝜑𝑢 ∈ ran 𝐹) → ((𝐹 ↾ ran 𝐺)‘(𝐺𝑢)) = (𝐹‘(𝐺𝑢)))
167 simpr 485 . . . . . . . . . . . 12 ((𝜑𝑢 ∈ ran 𝐹) → 𝑢 ∈ ran 𝐹)
168159a1i 11 . . . . . . . . . . . 12 ((𝜑𝑢 ∈ ran 𝐹) → (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣) ∈ V)
16921, 34, 167, 168fvmptd3 6971 . . . . . . . . . . 11 ((𝜑𝑢 ∈ ran 𝐹) → (𝐺𝑢) = (𝑣 ∈ (𝐹 “ {𝑢})∀𝑡 ∈ (𝐹 “ {𝑢}) ¬ 𝑡𝑅𝑣))
170169, 18eqeltrd 2838 . . . . . . . . . 10 ((𝜑𝑢 ∈ ran 𝐹) → (𝐺𝑢) ∈ (𝐹 “ {𝑢}))
171 fvex 6855 . . . . . . . . . . . 12 (𝐺𝑢) ∈ V
172 eleq1 2825 . . . . . . . . . . . . . 14 (𝑤 = (𝐺𝑢) → (𝑤 ∈ (𝐹 “ {𝑢}) ↔ (𝐺𝑢) ∈ (𝐹 “ {𝑢})))
173 eleq1 2825 . . . . . . . . . . . . . . 15 (𝑤 = (𝐺𝑢) → (𝑤𝐴 ↔ (𝐺𝑢) ∈ 𝐴))
174 fveqeq2 6851 . . . . . . . . . . . . . . 15 (𝑤 = (𝐺𝑢) → ((𝐹𝑤) = 𝑢 ↔ (𝐹‘(𝐺𝑢)) = 𝑢))
175173, 174anbi12d 631 . . . . . . . . . . . . . 14 (𝑤 = (𝐺𝑢) → ((𝑤𝐴 ∧ (𝐹𝑤) = 𝑢) ↔ ((𝐺𝑢) ∈ 𝐴 ∧ (𝐹‘(𝐺𝑢)) = 𝑢)))
176172, 175bibi12d 345 . . . . . . . . . . . . 13 (𝑤 = (𝐺𝑢) → ((𝑤 ∈ (𝐹 “ {𝑢}) ↔ (𝑤𝐴 ∧ (𝐹𝑤) = 𝑢)) ↔ ((𝐺𝑢) ∈ (𝐹 “ {𝑢}) ↔ ((𝐺𝑢) ∈ 𝐴 ∧ (𝐹‘(𝐺𝑢)) = 𝑢))))
177176imbi2d 340 . . . . . . . . . . . 12 (𝑤 = (𝐺𝑢) → ((𝜑 → (𝑤 ∈ (𝐹 “ {𝑢}) ↔ (𝑤𝐴 ∧ (𝐹𝑤) = 𝑢))) ↔ (𝜑 → ((𝐺𝑢) ∈ (𝐹 “ {𝑢}) ↔ ((𝐺𝑢) ∈ 𝐴 ∧ (𝐹‘(𝐺𝑢)) = 𝑢)))))
1783, 122syl 17 . . . . . . . . . . . 12 (𝜑 → (𝑤 ∈ (𝐹 “ {𝑢}) ↔ (𝑤𝐴 ∧ (𝐹𝑤) = 𝑢)))
179171, 177, 178vtocl 3518 . . . . . . . . . . 11 (𝜑 → ((𝐺𝑢) ∈ (𝐹 “ {𝑢}) ↔ ((𝐺𝑢) ∈ 𝐴 ∧ (𝐹‘(𝐺𝑢)) = 𝑢)))
180179adantr 481 . . . . . . . . . 10 ((𝜑𝑢 ∈ ran 𝐹) → ((𝐺𝑢) ∈ (𝐹 “ {𝑢}) ↔ ((𝐺𝑢) ∈ 𝐴 ∧ (𝐹‘(𝐺𝑢)) = 𝑢)))
181170, 180mpbid 231 . . . . . . . . 9 ((𝜑𝑢 ∈ ran 𝐹) → ((𝐺𝑢) ∈ 𝐴 ∧ (𝐹‘(𝐺𝑢)) = 𝑢))
182181simprd 496 . . . . . . . 8 ((𝜑𝑢 ∈ ran 𝐹) → (𝐹‘(𝐺𝑢)) = 𝑢)
183166, 182eqtr2d 2777 . . . . . . 7 ((𝜑𝑢 ∈ ran 𝐹) → 𝑢 = ((𝐹 ↾ ran 𝐺)‘(𝐺𝑢)))
184 fveq2 6842 . . . . . . . 8 (𝑤 = (𝐺𝑢) → ((𝐹 ↾ ran 𝐺)‘𝑤) = ((𝐹 ↾ ran 𝐺)‘(𝐺𝑢)))
185184rspceeqv 3595 . . . . . . 7 (((𝐺𝑢) ∈ ran 𝐺𝑢 = ((𝐹 ↾ ran 𝐺)‘(𝐺𝑢))) → ∃𝑤 ∈ ran 𝐺 𝑢 = ((𝐹 ↾ ran 𝐺)‘𝑤))
186165, 183, 185syl2anc 584 . . . . . 6 ((𝜑𝑢 ∈ ran 𝐹) → ∃𝑤 ∈ ran 𝐺 𝑢 = ((𝐹 ↾ ran 𝐺)‘𝑤))
187186ralrimiva 3143 . . . . 5 (𝜑 → ∀𝑢 ∈ ran 𝐹𝑤 ∈ ran 𝐺 𝑢 = ((𝐹 ↾ ran 𝐺)‘𝑤))
188 dffo3 7052 . . . . 5 ((𝐹 ↾ ran 𝐺):ran 𝐺onto→ran 𝐹 ↔ ((𝐹 ↾ ran 𝐺):ran 𝐺⟶ran 𝐹 ∧ ∀𝑢 ∈ ran 𝐹𝑤 ∈ ran 𝐺 𝑢 = ((𝐹 ↾ ran 𝐺)‘𝑤)))
18942, 187, 188sylanbrc 583 . . . 4 (𝜑 → (𝐹 ↾ ran 𝐺):ran 𝐺onto→ran 𝐹)
190 df-f1o 6503 . . . 4 ((𝐹 ↾ ran 𝐺):ran 𝐺1-1-onto→ran 𝐹 ↔ ((𝐹 ↾ ran 𝐺):ran 𝐺1-1→ran 𝐹 ∧ (𝐹 ↾ ran 𝐺):ran 𝐺onto→ran 𝐹))
191158, 189, 190sylanbrc 583 . . 3 (𝜑 → (𝐹 ↾ ran 𝐺):ran 𝐺1-1-onto→ran 𝐹)
192 reseq2 5932 . . . . 5 (𝑣 = ran 𝐺 → (𝐹𝑣) = (𝐹 ↾ ran 𝐺))
193 id 22 . . . . 5 (𝑣 = ran 𝐺𝑣 = ran 𝐺)
194 eqidd 2737 . . . . 5 (𝑣 = ran 𝐺 → ran 𝐹 = ran 𝐹)
195192, 193, 194f1oeq123d 6778 . . . 4 (𝑣 = ran 𝐺 → ((𝐹𝑣):𝑣1-1-onto→ran 𝐹 ↔ (𝐹 ↾ ran 𝐺):ran 𝐺1-1-onto→ran 𝐹))
196195rspcev 3581 . . 3 ((ran 𝐺 ∈ 𝒫 𝐴 ∧ (𝐹 ↾ ran 𝐺):ran 𝐺1-1-onto→ran 𝐹) → ∃𝑣 ∈ 𝒫 𝐴(𝐹𝑣):𝑣1-1-onto→ran 𝐹)
19739, 191, 196syl2anc 584 . 2 (𝜑 → ∃𝑣 ∈ 𝒫 𝐴(𝐹𝑣):𝑣1-1-onto→ran 𝐹)
198 reseq2 5932 . . . 4 (𝑣 = 𝑥 → (𝐹𝑣) = (𝐹𝑥))
199 id 22 . . . 4 (𝑣 = 𝑥𝑣 = 𝑥)
200 eqidd 2737 . . . 4 (𝑣 = 𝑥 → ran 𝐹 = ran 𝐹)
201198, 199, 200f1oeq123d 6778 . . 3 (𝑣 = 𝑥 → ((𝐹𝑣):𝑣1-1-onto→ran 𝐹 ↔ (𝐹𝑥):𝑥1-1-onto→ran 𝐹))
202201cbvrexvw 3226 . 2 (∃𝑣 ∈ 𝒫 𝐴(𝐹𝑣):𝑣1-1-onto→ran 𝐹 ↔ ∃𝑥 ∈ 𝒫 𝐴(𝐹𝑥):𝑥1-1-onto→ran 𝐹)
203197, 202sylib 217 1 (𝜑 → ∃𝑥 ∈ 𝒫 𝐴(𝐹𝑥):𝑥1-1-onto→ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  ∃!wreu 3351  Vcvv 3445  wss 3910  c0 4282  𝒫 cpw 4560  {csn 4586   class class class wbr 5105  cmpt 5188   Or wor 5544   We wwe 5587  ccnv 5632  dom cdm 5633  ran crn 5634  cres 5635  cima 5636   Fn wfn 6491  wf 6492  1-1wf1 6493  ontowfo 6494  1-1-ontowf1o 6495  cfv 6496  crio 7312
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-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-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  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-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313
This theorem is referenced by:  wessf1orn  43394
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