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Theorem fnwelem 7972
Description: Lemma for fnwe 7973. (Contributed by Mario Carneiro, 10-Mar-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)
Hypotheses
Ref Expression
fnwe.1 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)))}
fnwe.2 (𝜑𝐹:𝐴𝐵)
fnwe.3 (𝜑𝑅 We 𝐵)
fnwe.4 (𝜑𝑆 We 𝐴)
fnwe.5 (𝜑 → (𝐹𝑤) ∈ V)
fnwelem.6 𝑄 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ (𝐵 × 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ∧ ((1st𝑢)𝑅(1st𝑣) ∨ ((1st𝑢) = (1st𝑣) ∧ (2nd𝑢)𝑆(2nd𝑣))))}
fnwelem.7 𝐺 = (𝑧𝐴 ↦ ⟨(𝐹𝑧), 𝑧⟩)
Assertion
Ref Expression
fnwelem (𝜑𝑇 We 𝐴)
Distinct variable groups:   𝑣,𝑢,𝑤,𝑥,𝑦,𝑧,𝐴   𝑢,𝐵,𝑣,𝑤,𝑥,𝑦,𝑧   𝑤,𝐺,𝑥,𝑦   𝜑,𝑤,𝑥,𝑧   𝑢,𝐹,𝑣,𝑤,𝑥,𝑦,𝑧   𝑤,𝑄,𝑥,𝑦   𝑢,𝑅,𝑣,𝑤,𝑥,𝑦   𝑢,𝑆,𝑣,𝑤,𝑥,𝑦   𝑤,𝑇
Allowed substitution hints:   𝜑(𝑦,𝑣,𝑢)   𝑄(𝑧,𝑣,𝑢)   𝑅(𝑧)   𝑆(𝑧)   𝑇(𝑥,𝑦,𝑧,𝑣,𝑢)   𝐺(𝑧,𝑣,𝑢)

Proof of Theorem fnwelem
StepHypRef Expression
1 fnwe.2 . . . 4 (𝜑𝐹:𝐴𝐵)
2 ffvelrn 6959 . . . . . 6 ((𝐹:𝐴𝐵𝑧𝐴) → (𝐹𝑧) ∈ 𝐵)
3 simpr 485 . . . . . 6 ((𝐹:𝐴𝐵𝑧𝐴) → 𝑧𝐴)
42, 3opelxpd 5627 . . . . 5 ((𝐹:𝐴𝐵𝑧𝐴) → ⟨(𝐹𝑧), 𝑧⟩ ∈ (𝐵 × 𝐴))
5 fnwelem.7 . . . . 5 𝐺 = (𝑧𝐴 ↦ ⟨(𝐹𝑧), 𝑧⟩)
64, 5fmptd 6988 . . . 4 (𝐹:𝐴𝐵𝐺:𝐴⟶(𝐵 × 𝐴))
7 frn 6607 . . . 4 (𝐺:𝐴⟶(𝐵 × 𝐴) → ran 𝐺 ⊆ (𝐵 × 𝐴))
81, 6, 73syl 18 . . 3 (𝜑 → ran 𝐺 ⊆ (𝐵 × 𝐴))
9 fnwe.3 . . . 4 (𝜑𝑅 We 𝐵)
10 fnwe.4 . . . 4 (𝜑𝑆 We 𝐴)
11 fnwelem.6 . . . . 5 𝑄 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ (𝐵 × 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ∧ ((1st𝑢)𝑅(1st𝑣) ∨ ((1st𝑢) = (1st𝑣) ∧ (2nd𝑢)𝑆(2nd𝑣))))}
1211wexp 7971 . . . 4 ((𝑅 We 𝐵𝑆 We 𝐴) → 𝑄 We (𝐵 × 𝐴))
139, 10, 12syl2anc 584 . . 3 (𝜑𝑄 We (𝐵 × 𝐴))
14 wess 5576 . . 3 (ran 𝐺 ⊆ (𝐵 × 𝐴) → (𝑄 We (𝐵 × 𝐴) → 𝑄 We ran 𝐺))
158, 13, 14sylc 65 . 2 (𝜑𝑄 We ran 𝐺)
16 fveq2 6774 . . . . . . . . . . . 12 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
17 id 22 . . . . . . . . . . . 12 (𝑧 = 𝑥𝑧 = 𝑥)
1816, 17opeq12d 4812 . . . . . . . . . . 11 (𝑧 = 𝑥 → ⟨(𝐹𝑧), 𝑧⟩ = ⟨(𝐹𝑥), 𝑥⟩)
19 opex 5379 . . . . . . . . . . 11 ⟨(𝐹𝑥), 𝑥⟩ ∈ V
2018, 5, 19fvmpt 6875 . . . . . . . . . 10 (𝑥𝐴 → (𝐺𝑥) = ⟨(𝐹𝑥), 𝑥⟩)
21 fveq2 6774 . . . . . . . . . . . 12 (𝑧 = 𝑦 → (𝐹𝑧) = (𝐹𝑦))
22 id 22 . . . . . . . . . . . 12 (𝑧 = 𝑦𝑧 = 𝑦)
2321, 22opeq12d 4812 . . . . . . . . . . 11 (𝑧 = 𝑦 → ⟨(𝐹𝑧), 𝑧⟩ = ⟨(𝐹𝑦), 𝑦⟩)
24 opex 5379 . . . . . . . . . . 11 ⟨(𝐹𝑦), 𝑦⟩ ∈ V
2523, 5, 24fvmpt 6875 . . . . . . . . . 10 (𝑦𝐴 → (𝐺𝑦) = ⟨(𝐹𝑦), 𝑦⟩)
2620, 25eqeqan12d 2752 . . . . . . . . 9 ((𝑥𝐴𝑦𝐴) → ((𝐺𝑥) = (𝐺𝑦) ↔ ⟨(𝐹𝑥), 𝑥⟩ = ⟨(𝐹𝑦), 𝑦⟩))
27 fvex 6787 . . . . . . . . . . 11 (𝐹𝑥) ∈ V
28 vex 3436 . . . . . . . . . . 11 𝑥 ∈ V
2927, 28opth 5391 . . . . . . . . . 10 (⟨(𝐹𝑥), 𝑥⟩ = ⟨(𝐹𝑦), 𝑦⟩ ↔ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 = 𝑦))
3029simprbi 497 . . . . . . . . 9 (⟨(𝐹𝑥), 𝑥⟩ = ⟨(𝐹𝑦), 𝑦⟩ → 𝑥 = 𝑦)
3126, 30syl6bi 252 . . . . . . . 8 ((𝑥𝐴𝑦𝐴) → ((𝐺𝑥) = (𝐺𝑦) → 𝑥 = 𝑦))
3231rgen2 3120 . . . . . . 7 𝑥𝐴𝑦𝐴 ((𝐺𝑥) = (𝐺𝑦) → 𝑥 = 𝑦)
33 dff13 7128 . . . . . . 7 (𝐺:𝐴1-1→(𝐵 × 𝐴) ↔ (𝐺:𝐴⟶(𝐵 × 𝐴) ∧ ∀𝑥𝐴𝑦𝐴 ((𝐺𝑥) = (𝐺𝑦) → 𝑥 = 𝑦)))
346, 32, 33sylanblrc 590 . . . . . 6 (𝐹:𝐴𝐵𝐺:𝐴1-1→(𝐵 × 𝐴))
35 f1f1orn 6727 . . . . . 6 (𝐺:𝐴1-1→(𝐵 × 𝐴) → 𝐺:𝐴1-1-onto→ran 𝐺)
36 f1ocnv 6728 . . . . . 6 (𝐺:𝐴1-1-onto→ran 𝐺𝐺:ran 𝐺1-1-onto𝐴)
371, 34, 35, 364syl 19 . . . . 5 (𝜑𝐺:ran 𝐺1-1-onto𝐴)
38 eqid 2738 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ (𝐺𝑥)𝑄(𝐺𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ (𝐺𝑥)𝑄(𝐺𝑦))}
3938f1oiso2 7223 . . . . . 6 (𝐺:ran 𝐺1-1-onto𝐴𝐺 Isom 𝑄, {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ (𝐺𝑥)𝑄(𝐺𝑦))} (ran 𝐺, 𝐴))
40 fnwe.1 . . . . . . . 8 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)))}
41 frel 6605 . . . . . . . . . . . . . . . 16 (𝐺:𝐴⟶(𝐵 × 𝐴) → Rel 𝐺)
42 dfrel2 6092 . . . . . . . . . . . . . . . 16 (Rel 𝐺𝐺 = 𝐺)
4341, 42sylib 217 . . . . . . . . . . . . . . 15 (𝐺:𝐴⟶(𝐵 × 𝐴) → 𝐺 = 𝐺)
4443fveq1d 6776 . . . . . . . . . . . . . 14 (𝐺:𝐴⟶(𝐵 × 𝐴) → (𝐺𝑥) = (𝐺𝑥))
4543fveq1d 6776 . . . . . . . . . . . . . 14 (𝐺:𝐴⟶(𝐵 × 𝐴) → (𝐺𝑦) = (𝐺𝑦))
4644, 45breq12d 5087 . . . . . . . . . . . . 13 (𝐺:𝐴⟶(𝐵 × 𝐴) → ((𝐺𝑥)𝑄(𝐺𝑦) ↔ (𝐺𝑥)𝑄(𝐺𝑦)))
476, 46syl 17 . . . . . . . . . . . 12 (𝐹:𝐴𝐵 → ((𝐺𝑥)𝑄(𝐺𝑦) ↔ (𝐺𝑥)𝑄(𝐺𝑦)))
4847adantr 481 . . . . . . . . . . 11 ((𝐹:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐺𝑥)𝑄(𝐺𝑦) ↔ (𝐺𝑥)𝑄(𝐺𝑦)))
4920, 25breqan12d 5090 . . . . . . . . . . . 12 ((𝑥𝐴𝑦𝐴) → ((𝐺𝑥)𝑄(𝐺𝑦) ↔ ⟨(𝐹𝑥), 𝑥𝑄⟨(𝐹𝑦), 𝑦⟩))
5049adantl 482 . . . . . . . . . . 11 ((𝐹:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐺𝑥)𝑄(𝐺𝑦) ↔ ⟨(𝐹𝑥), 𝑥𝑄⟨(𝐹𝑦), 𝑦⟩))
51 eleq1 2826 . . . . . . . . . . . . . . . 16 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → (𝑢 ∈ (𝐵 × 𝐴) ↔ ⟨(𝐹𝑥), 𝑥⟩ ∈ (𝐵 × 𝐴)))
52 opelxp 5625 . . . . . . . . . . . . . . . 16 (⟨(𝐹𝑥), 𝑥⟩ ∈ (𝐵 × 𝐴) ↔ ((𝐹𝑥) ∈ 𝐵𝑥𝐴))
5351, 52bitrdi 287 . . . . . . . . . . . . . . 15 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → (𝑢 ∈ (𝐵 × 𝐴) ↔ ((𝐹𝑥) ∈ 𝐵𝑥𝐴)))
5453anbi1d 630 . . . . . . . . . . . . . 14 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → ((𝑢 ∈ (𝐵 × 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ↔ (((𝐹𝑥) ∈ 𝐵𝑥𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴))))
5527, 28op1std 7841 . . . . . . . . . . . . . . . 16 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → (1st𝑢) = (𝐹𝑥))
5655breq1d 5084 . . . . . . . . . . . . . . 15 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → ((1st𝑢)𝑅(1st𝑣) ↔ (𝐹𝑥)𝑅(1st𝑣)))
5755eqeq1d 2740 . . . . . . . . . . . . . . . 16 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → ((1st𝑢) = (1st𝑣) ↔ (𝐹𝑥) = (1st𝑣)))
5827, 28op2ndd 7842 . . . . . . . . . . . . . . . . 17 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → (2nd𝑢) = 𝑥)
5958breq1d 5084 . . . . . . . . . . . . . . . 16 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → ((2nd𝑢)𝑆(2nd𝑣) ↔ 𝑥𝑆(2nd𝑣)))
6057, 59anbi12d 631 . . . . . . . . . . . . . . 15 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → (((1st𝑢) = (1st𝑣) ∧ (2nd𝑢)𝑆(2nd𝑣)) ↔ ((𝐹𝑥) = (1st𝑣) ∧ 𝑥𝑆(2nd𝑣))))
6156, 60orbi12d 916 . . . . . . . . . . . . . 14 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → (((1st𝑢)𝑅(1st𝑣) ∨ ((1st𝑢) = (1st𝑣) ∧ (2nd𝑢)𝑆(2nd𝑣))) ↔ ((𝐹𝑥)𝑅(1st𝑣) ∨ ((𝐹𝑥) = (1st𝑣) ∧ 𝑥𝑆(2nd𝑣)))))
6254, 61anbi12d 631 . . . . . . . . . . . . 13 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → (((𝑢 ∈ (𝐵 × 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ∧ ((1st𝑢)𝑅(1st𝑣) ∨ ((1st𝑢) = (1st𝑣) ∧ (2nd𝑢)𝑆(2nd𝑣)))) ↔ ((((𝐹𝑥) ∈ 𝐵𝑥𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ∧ ((𝐹𝑥)𝑅(1st𝑣) ∨ ((𝐹𝑥) = (1st𝑣) ∧ 𝑥𝑆(2nd𝑣))))))
63 eleq1 2826 . . . . . . . . . . . . . . . 16 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → (𝑣 ∈ (𝐵 × 𝐴) ↔ ⟨(𝐹𝑦), 𝑦⟩ ∈ (𝐵 × 𝐴)))
64 opelxp 5625 . . . . . . . . . . . . . . . 16 (⟨(𝐹𝑦), 𝑦⟩ ∈ (𝐵 × 𝐴) ↔ ((𝐹𝑦) ∈ 𝐵𝑦𝐴))
6563, 64bitrdi 287 . . . . . . . . . . . . . . 15 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → (𝑣 ∈ (𝐵 × 𝐴) ↔ ((𝐹𝑦) ∈ 𝐵𝑦𝐴)))
6665anbi2d 629 . . . . . . . . . . . . . 14 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → ((((𝐹𝑥) ∈ 𝐵𝑥𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ↔ (((𝐹𝑥) ∈ 𝐵𝑥𝐴) ∧ ((𝐹𝑦) ∈ 𝐵𝑦𝐴))))
67 fvex 6787 . . . . . . . . . . . . . . . . 17 (𝐹𝑦) ∈ V
68 vex 3436 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
6967, 68op1std 7841 . . . . . . . . . . . . . . . 16 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → (1st𝑣) = (𝐹𝑦))
7069breq2d 5086 . . . . . . . . . . . . . . 15 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → ((𝐹𝑥)𝑅(1st𝑣) ↔ (𝐹𝑥)𝑅(𝐹𝑦)))
7169eqeq2d 2749 . . . . . . . . . . . . . . . 16 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → ((𝐹𝑥) = (1st𝑣) ↔ (𝐹𝑥) = (𝐹𝑦)))
7267, 68op2ndd 7842 . . . . . . . . . . . . . . . . 17 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → (2nd𝑣) = 𝑦)
7372breq2d 5086 . . . . . . . . . . . . . . . 16 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → (𝑥𝑆(2nd𝑣) ↔ 𝑥𝑆𝑦))
7471, 73anbi12d 631 . . . . . . . . . . . . . . 15 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → (((𝐹𝑥) = (1st𝑣) ∧ 𝑥𝑆(2nd𝑣)) ↔ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)))
7570, 74orbi12d 916 . . . . . . . . . . . . . 14 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → (((𝐹𝑥)𝑅(1st𝑣) ∨ ((𝐹𝑥) = (1st𝑣) ∧ 𝑥𝑆(2nd𝑣))) ↔ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦))))
7666, 75anbi12d 631 . . . . . . . . . . . . 13 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → (((((𝐹𝑥) ∈ 𝐵𝑥𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ∧ ((𝐹𝑥)𝑅(1st𝑣) ∨ ((𝐹𝑥) = (1st𝑣) ∧ 𝑥𝑆(2nd𝑣)))) ↔ ((((𝐹𝑥) ∈ 𝐵𝑥𝐴) ∧ ((𝐹𝑦) ∈ 𝐵𝑦𝐴)) ∧ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)))))
7719, 24, 62, 76, 11brab 5456 . . . . . . . . . . . 12 (⟨(𝐹𝑥), 𝑥𝑄⟨(𝐹𝑦), 𝑦⟩ ↔ ((((𝐹𝑥) ∈ 𝐵𝑥𝐴) ∧ ((𝐹𝑦) ∈ 𝐵𝑦𝐴)) ∧ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦))))
78 ffvelrn 6959 . . . . . . . . . . . . . . 15 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
79 simpr 485 . . . . . . . . . . . . . . 15 ((𝐹:𝐴𝐵𝑥𝐴) → 𝑥𝐴)
8078, 79jca 512 . . . . . . . . . . . . . 14 ((𝐹:𝐴𝐵𝑥𝐴) → ((𝐹𝑥) ∈ 𝐵𝑥𝐴))
81 ffvelrn 6959 . . . . . . . . . . . . . . 15 ((𝐹:𝐴𝐵𝑦𝐴) → (𝐹𝑦) ∈ 𝐵)
82 simpr 485 . . . . . . . . . . . . . . 15 ((𝐹:𝐴𝐵𝑦𝐴) → 𝑦𝐴)
8381, 82jca 512 . . . . . . . . . . . . . 14 ((𝐹:𝐴𝐵𝑦𝐴) → ((𝐹𝑦) ∈ 𝐵𝑦𝐴))
8480, 83anim12dan 619 . . . . . . . . . . . . 13 ((𝐹:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (((𝐹𝑥) ∈ 𝐵𝑥𝐴) ∧ ((𝐹𝑦) ∈ 𝐵𝑦𝐴)))
8584biantrurd 533 . . . . . . . . . . . 12 ((𝐹:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)) ↔ ((((𝐹𝑥) ∈ 𝐵𝑥𝐴) ∧ ((𝐹𝑦) ∈ 𝐵𝑦𝐴)) ∧ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)))))
8677, 85bitr4id 290 . . . . . . . . . . 11 ((𝐹:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (⟨(𝐹𝑥), 𝑥𝑄⟨(𝐹𝑦), 𝑦⟩ ↔ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦))))
8748, 50, 863bitrrd 306 . . . . . . . . . 10 ((𝐹:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)) ↔ (𝐺𝑥)𝑄(𝐺𝑦)))
8887pm5.32da 579 . . . . . . . . 9 (𝐹:𝐴𝐵 → (((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦))) ↔ ((𝑥𝐴𝑦𝐴) ∧ (𝐺𝑥)𝑄(𝐺𝑦))))
8988opabbidv 5140 . . . . . . . 8 (𝐹:𝐴𝐵 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ (𝐺𝑥)𝑄(𝐺𝑦))})
9040, 89eqtrid 2790 . . . . . . 7 (𝐹:𝐴𝐵𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ (𝐺𝑥)𝑄(𝐺𝑦))})
91 isoeq3 7190 . . . . . . 7 (𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ (𝐺𝑥)𝑄(𝐺𝑦))} → (𝐺 Isom 𝑄, 𝑇 (ran 𝐺, 𝐴) ↔ 𝐺 Isom 𝑄, {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ (𝐺𝑥)𝑄(𝐺𝑦))} (ran 𝐺, 𝐴)))
9290, 91syl 17 . . . . . 6 (𝐹:𝐴𝐵 → (𝐺 Isom 𝑄, 𝑇 (ran 𝐺, 𝐴) ↔ 𝐺 Isom 𝑄, {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ (𝐺𝑥)𝑄(𝐺𝑦))} (ran 𝐺, 𝐴)))
9339, 92syl5ibr 245 . . . . 5 (𝐹:𝐴𝐵 → (𝐺:ran 𝐺1-1-onto𝐴𝐺 Isom 𝑄, 𝑇 (ran 𝐺, 𝐴)))
941, 37, 93sylc 65 . . . 4 (𝜑𝐺 Isom 𝑄, 𝑇 (ran 𝐺, 𝐴))
95 isocnv 7201 . . . 4 (𝐺 Isom 𝑄, 𝑇 (ran 𝐺, 𝐴) → 𝐺 Isom 𝑇, 𝑄 (𝐴, ran 𝐺))
9694, 95syl 17 . . 3 (𝜑𝐺 Isom 𝑇, 𝑄 (𝐴, ran 𝐺))
97 imacnvcnv 6109 . . . . 5 (𝐺𝑤) = (𝐺𝑤)
98 fnwe.5 . . . . . . 7 (𝜑 → (𝐹𝑤) ∈ V)
99 vex 3436 . . . . . . 7 𝑤 ∈ V
100 xpexg 7600 . . . . . . 7 (((𝐹𝑤) ∈ V ∧ 𝑤 ∈ V) → ((𝐹𝑤) × 𝑤) ∈ V)
10198, 99, 100sylancl 586 . . . . . 6 (𝜑 → ((𝐹𝑤) × 𝑤) ∈ V)
102 imadmres 6137 . . . . . . 7 (𝐺 “ dom (𝐺𝑤)) = (𝐺𝑤)
103 dmres 5913 . . . . . . . . . . 11 dom (𝐺𝑤) = (𝑤 ∩ dom 𝐺)
104103elin2 4131 . . . . . . . . . 10 (𝑥 ∈ dom (𝐺𝑤) ↔ (𝑥𝑤𝑥 ∈ dom 𝐺))
105 simprr 770 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → 𝑥 ∈ dom 𝐺)
106 f1dm 6674 . . . . . . . . . . . . . . 15 (𝐺:𝐴1-1→(𝐵 × 𝐴) → dom 𝐺 = 𝐴)
1071, 34, 1063syl 18 . . . . . . . . . . . . . 14 (𝜑 → dom 𝐺 = 𝐴)
108107adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → dom 𝐺 = 𝐴)
109105, 108eleqtrd 2841 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → 𝑥𝐴)
110109, 20syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → (𝐺𝑥) = ⟨(𝐹𝑥), 𝑥⟩)
1111ffnd 6601 . . . . . . . . . . . . . . 15 (𝜑𝐹 Fn 𝐴)
112111adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → 𝐹 Fn 𝐴)
113 dmres 5913 . . . . . . . . . . . . . . 15 dom (𝐹𝑤) = (𝑤 ∩ dom 𝐹)
114 inss2 4163 . . . . . . . . . . . . . . . 16 (𝑤 ∩ dom 𝐹) ⊆ dom 𝐹
115112fndmd 6538 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → dom 𝐹 = 𝐴)
116114, 115sseqtrid 3973 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → (𝑤 ∩ dom 𝐹) ⊆ 𝐴)
117113, 116eqsstrid 3969 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → dom (𝐹𝑤) ⊆ 𝐴)
118 simprl 768 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → 𝑥𝑤)
119109, 115eleqtrrd 2842 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → 𝑥 ∈ dom 𝐹)
120113elin2 4131 . . . . . . . . . . . . . . 15 (𝑥 ∈ dom (𝐹𝑤) ↔ (𝑥𝑤𝑥 ∈ dom 𝐹))
121118, 119, 120sylanbrc 583 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → 𝑥 ∈ dom (𝐹𝑤))
122 fnfvima 7109 . . . . . . . . . . . . . 14 ((𝐹 Fn 𝐴 ∧ dom (𝐹𝑤) ⊆ 𝐴𝑥 ∈ dom (𝐹𝑤)) → (𝐹𝑥) ∈ (𝐹 “ dom (𝐹𝑤)))
123112, 117, 121, 122syl3anc 1370 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → (𝐹𝑥) ∈ (𝐹 “ dom (𝐹𝑤)))
124 imadmres 6137 . . . . . . . . . . . . 13 (𝐹 “ dom (𝐹𝑤)) = (𝐹𝑤)
125123, 124eleqtrdi 2849 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → (𝐹𝑥) ∈ (𝐹𝑤))
126125, 118opelxpd 5627 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → ⟨(𝐹𝑥), 𝑥⟩ ∈ ((𝐹𝑤) × 𝑤))
127110, 126eqeltrd 2839 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → (𝐺𝑥) ∈ ((𝐹𝑤) × 𝑤))
128104, 127sylan2b 594 . . . . . . . . 9 ((𝜑𝑥 ∈ dom (𝐺𝑤)) → (𝐺𝑥) ∈ ((𝐹𝑤) × 𝑤))
129128ralrimiva 3103 . . . . . . . 8 (𝜑 → ∀𝑥 ∈ dom (𝐺𝑤)(𝐺𝑥) ∈ ((𝐹𝑤) × 𝑤))
130 f1fun 6672 . . . . . . . . . 10 (𝐺:𝐴1-1→(𝐵 × 𝐴) → Fun 𝐺)
1311, 34, 1303syl 18 . . . . . . . . 9 (𝜑 → Fun 𝐺)
132 resss 5916 . . . . . . . . . 10 (𝐺𝑤) ⊆ 𝐺
133 dmss 5811 . . . . . . . . . 10 ((𝐺𝑤) ⊆ 𝐺 → dom (𝐺𝑤) ⊆ dom 𝐺)
134132, 133ax-mp 5 . . . . . . . . 9 dom (𝐺𝑤) ⊆ dom 𝐺
135 funimass4 6834 . . . . . . . . 9 ((Fun 𝐺 ∧ dom (𝐺𝑤) ⊆ dom 𝐺) → ((𝐺 “ dom (𝐺𝑤)) ⊆ ((𝐹𝑤) × 𝑤) ↔ ∀𝑥 ∈ dom (𝐺𝑤)(𝐺𝑥) ∈ ((𝐹𝑤) × 𝑤)))
136131, 134, 135sylancl 586 . . . . . . . 8 (𝜑 → ((𝐺 “ dom (𝐺𝑤)) ⊆ ((𝐹𝑤) × 𝑤) ↔ ∀𝑥 ∈ dom (𝐺𝑤)(𝐺𝑥) ∈ ((𝐹𝑤) × 𝑤)))
137129, 136mpbird 256 . . . . . . 7 (𝜑 → (𝐺 “ dom (𝐺𝑤)) ⊆ ((𝐹𝑤) × 𝑤))
138102, 137eqsstrrid 3970 . . . . . 6 (𝜑 → (𝐺𝑤) ⊆ ((𝐹𝑤) × 𝑤))
139101, 138ssexd 5248 . . . . 5 (𝜑 → (𝐺𝑤) ∈ V)
14097, 139eqeltrid 2843 . . . 4 (𝜑 → (𝐺𝑤) ∈ V)
141140alrimiv 1930 . . 3 (𝜑 → ∀𝑤(𝐺𝑤) ∈ V)
142 isowe2 7221 . . 3 ((𝐺 Isom 𝑇, 𝑄 (𝐴, ran 𝐺) ∧ ∀𝑤(𝐺𝑤) ∈ V) → (𝑄 We ran 𝐺𝑇 We 𝐴))
14396, 141, 142syl2anc 584 . 2 (𝜑 → (𝑄 We ran 𝐺𝑇 We 𝐴))
14415, 143mpd 15 1 (𝜑𝑇 We 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 844  wal 1537   = wceq 1539  wcel 2106  wral 3064  Vcvv 3432  cin 3886  wss 3887  cop 4567   class class class wbr 5074  {copab 5136  cmpt 5157   We wwe 5543   × cxp 5587  ccnv 5588  dom cdm 5589  ran crn 5590  cres 5591  cima 5592  Rel wrel 5594  Fun wfun 6427   Fn wfn 6428  wf 6429  1-1wf1 6430  1-1-ontowf1o 6432  cfv 6433   Isom wiso 6434  1st c1st 7829  2nd c2nd 7830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-1st 7831  df-2nd 7832
This theorem is referenced by:  fnwe  7973
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