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Theorem fnwelem 7814
Description: Lemma for fnwe 7815. (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 6841 . . . . . 6 ((𝐹:𝐴𝐵𝑧𝐴) → (𝐹𝑧) ∈ 𝐵)
3 simpr 485 . . . . . 6 ((𝐹:𝐴𝐵𝑧𝐴) → 𝑧𝐴)
42, 3opelxpd 5586 . . . . 5 ((𝐹:𝐴𝐵𝑧𝐴) → ⟨(𝐹𝑧), 𝑧⟩ ∈ (𝐵 × 𝐴))
5 fnwelem.7 . . . . 5 𝐺 = (𝑧𝐴 ↦ ⟨(𝐹𝑧), 𝑧⟩)
64, 5fmptd 6870 . . . 4 (𝐹:𝐴𝐵𝐺:𝐴⟶(𝐵 × 𝐴))
7 frn 6513 . . . 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 7813 . . . 4 ((𝑅 We 𝐵𝑆 We 𝐴) → 𝑄 We (𝐵 × 𝐴))
139, 10, 12syl2anc 584 . . 3 (𝜑𝑄 We (𝐵 × 𝐴))
14 wess 5535 . . 3 (ran 𝐺 ⊆ (𝐵 × 𝐴) → (𝑄 We (𝐵 × 𝐴) → 𝑄 We ran 𝐺))
158, 13, 14sylc 65 . 2 (𝜑𝑄 We ran 𝐺)
16 fveq2 6663 . . . . . . . . . . . 12 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
17 id 22 . . . . . . . . . . . 12 (𝑧 = 𝑥𝑧 = 𝑥)
1816, 17opeq12d 4803 . . . . . . . . . . 11 (𝑧 = 𝑥 → ⟨(𝐹𝑧), 𝑧⟩ = ⟨(𝐹𝑥), 𝑥⟩)
19 opex 5347 . . . . . . . . . . 11 ⟨(𝐹𝑥), 𝑥⟩ ∈ V
2018, 5, 19fvmpt 6761 . . . . . . . . . 10 (𝑥𝐴 → (𝐺𝑥) = ⟨(𝐹𝑥), 𝑥⟩)
21 fveq2 6663 . . . . . . . . . . . 12 (𝑧 = 𝑦 → (𝐹𝑧) = (𝐹𝑦))
22 id 22 . . . . . . . . . . . 12 (𝑧 = 𝑦𝑧 = 𝑦)
2321, 22opeq12d 4803 . . . . . . . . . . 11 (𝑧 = 𝑦 → ⟨(𝐹𝑧), 𝑧⟩ = ⟨(𝐹𝑦), 𝑦⟩)
24 opex 5347 . . . . . . . . . . 11 ⟨(𝐹𝑦), 𝑦⟩ ∈ V
2523, 5, 24fvmpt 6761 . . . . . . . . . 10 (𝑦𝐴 → (𝐺𝑦) = ⟨(𝐹𝑦), 𝑦⟩)
2620, 25eqeqan12d 2835 . . . . . . . . 9 ((𝑥𝐴𝑦𝐴) → ((𝐺𝑥) = (𝐺𝑦) ↔ ⟨(𝐹𝑥), 𝑥⟩ = ⟨(𝐹𝑦), 𝑦⟩))
27 fvex 6676 . . . . . . . . . . 11 (𝐹𝑥) ∈ V
28 vex 3495 . . . . . . . . . . 11 𝑥 ∈ V
2927, 28opth 5359 . . . . . . . . . 10 (⟨(𝐹𝑥), 𝑥⟩ = ⟨(𝐹𝑦), 𝑦⟩ ↔ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 = 𝑦))
3029simprbi 497 . . . . . . . . 9 (⟨(𝐹𝑥), 𝑥⟩ = ⟨(𝐹𝑦), 𝑦⟩ → 𝑥 = 𝑦)
3126, 30syl6bi 254 . . . . . . . 8 ((𝑥𝐴𝑦𝐴) → ((𝐺𝑥) = (𝐺𝑦) → 𝑥 = 𝑦))
3231rgen2 3200 . . . . . . 7 𝑥𝐴𝑦𝐴 ((𝐺𝑥) = (𝐺𝑦) → 𝑥 = 𝑦)
33 dff13 7004 . . . . . . 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 6619 . . . . . 6 (𝐺:𝐴1-1→(𝐵 × 𝐴) → 𝐺:𝐴1-1-onto→ran 𝐺)
36 f1ocnv 6620 . . . . . 6 (𝐺:𝐴1-1-onto→ran 𝐺𝐺:ran 𝐺1-1-onto𝐴)
371, 34, 35, 364syl 19 . . . . 5 (𝜑𝐺:ran 𝐺1-1-onto𝐴)
38 eqid 2818 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ (𝐺𝑥)𝑄(𝐺𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ (𝐺𝑥)𝑄(𝐺𝑦))}
3938f1oiso2 7094 . . . . . 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 6512 . . . . . . . . . . . . . . . 16 (𝐺:𝐴⟶(𝐵 × 𝐴) → Rel 𝐺)
42 dfrel2 6039 . . . . . . . . . . . . . . . 16 (Rel 𝐺𝐺 = 𝐺)
4341, 42sylib 219 . . . . . . . . . . . . . . 15 (𝐺:𝐴⟶(𝐵 × 𝐴) → 𝐺 = 𝐺)
4443fveq1d 6665 . . . . . . . . . . . . . 14 (𝐺:𝐴⟶(𝐵 × 𝐴) → (𝐺𝑥) = (𝐺𝑥))
4543fveq1d 6665 . . . . . . . . . . . . . 14 (𝐺:𝐴⟶(𝐵 × 𝐴) → (𝐺𝑦) = (𝐺𝑦))
4644, 45breq12d 5070 . . . . . . . . . . . . 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 5073 . . . . . . . . . . . 12 ((𝑥𝐴𝑦𝐴) → ((𝐺𝑥)𝑄(𝐺𝑦) ↔ ⟨(𝐹𝑥), 𝑥𝑄⟨(𝐹𝑦), 𝑦⟩))
5049adantl 482 . . . . . . . . . . 11 ((𝐹:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐺𝑥)𝑄(𝐺𝑦) ↔ ⟨(𝐹𝑥), 𝑥𝑄⟨(𝐹𝑦), 𝑦⟩))
51 ffvelrn 6841 . . . . . . . . . . . . . . 15 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
52 simpr 485 . . . . . . . . . . . . . . 15 ((𝐹:𝐴𝐵𝑥𝐴) → 𝑥𝐴)
5351, 52jca 512 . . . . . . . . . . . . . 14 ((𝐹:𝐴𝐵𝑥𝐴) → ((𝐹𝑥) ∈ 𝐵𝑥𝐴))
54 ffvelrn 6841 . . . . . . . . . . . . . . 15 ((𝐹:𝐴𝐵𝑦𝐴) → (𝐹𝑦) ∈ 𝐵)
55 simpr 485 . . . . . . . . . . . . . . 15 ((𝐹:𝐴𝐵𝑦𝐴) → 𝑦𝐴)
5654, 55jca 512 . . . . . . . . . . . . . 14 ((𝐹:𝐴𝐵𝑦𝐴) → ((𝐹𝑦) ∈ 𝐵𝑦𝐴))
5753, 56anim12dan 618 . . . . . . . . . . . . 13 ((𝐹:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (((𝐹𝑥) ∈ 𝐵𝑥𝐴) ∧ ((𝐹𝑦) ∈ 𝐵𝑦𝐴)))
5857biantrurd 533 . . . . . . . . . . . 12 ((𝐹:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)) ↔ ((((𝐹𝑥) ∈ 𝐵𝑥𝐴) ∧ ((𝐹𝑦) ∈ 𝐵𝑦𝐴)) ∧ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)))))
59 eleq1 2897 . . . . . . . . . . . . . . . 16 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → (𝑢 ∈ (𝐵 × 𝐴) ↔ ⟨(𝐹𝑥), 𝑥⟩ ∈ (𝐵 × 𝐴)))
60 opelxp 5584 . . . . . . . . . . . . . . . 16 (⟨(𝐹𝑥), 𝑥⟩ ∈ (𝐵 × 𝐴) ↔ ((𝐹𝑥) ∈ 𝐵𝑥𝐴))
6159, 60syl6bb 288 . . . . . . . . . . . . . . 15 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → (𝑢 ∈ (𝐵 × 𝐴) ↔ ((𝐹𝑥) ∈ 𝐵𝑥𝐴)))
6261anbi1d 629 . . . . . . . . . . . . . 14 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → ((𝑢 ∈ (𝐵 × 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ↔ (((𝐹𝑥) ∈ 𝐵𝑥𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴))))
6327, 28op1std 7688 . . . . . . . . . . . . . . . 16 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → (1st𝑢) = (𝐹𝑥))
6463breq1d 5067 . . . . . . . . . . . . . . 15 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → ((1st𝑢)𝑅(1st𝑣) ↔ (𝐹𝑥)𝑅(1st𝑣)))
6563eqeq1d 2820 . . . . . . . . . . . . . . . 16 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → ((1st𝑢) = (1st𝑣) ↔ (𝐹𝑥) = (1st𝑣)))
6627, 28op2ndd 7689 . . . . . . . . . . . . . . . . 17 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → (2nd𝑢) = 𝑥)
6766breq1d 5067 . . . . . . . . . . . . . . . 16 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → ((2nd𝑢)𝑆(2nd𝑣) ↔ 𝑥𝑆(2nd𝑣)))
6865, 67anbi12d 630 . . . . . . . . . . . . . . 15 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → (((1st𝑢) = (1st𝑣) ∧ (2nd𝑢)𝑆(2nd𝑣)) ↔ ((𝐹𝑥) = (1st𝑣) ∧ 𝑥𝑆(2nd𝑣))))
6964, 68orbi12d 912 . . . . . . . . . . . . . 14 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → (((1st𝑢)𝑅(1st𝑣) ∨ ((1st𝑢) = (1st𝑣) ∧ (2nd𝑢)𝑆(2nd𝑣))) ↔ ((𝐹𝑥)𝑅(1st𝑣) ∨ ((𝐹𝑥) = (1st𝑣) ∧ 𝑥𝑆(2nd𝑣)))))
7062, 69anbi12d 630 . . . . . . . . . . . . 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))))))
71 eleq1 2897 . . . . . . . . . . . . . . . 16 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → (𝑣 ∈ (𝐵 × 𝐴) ↔ ⟨(𝐹𝑦), 𝑦⟩ ∈ (𝐵 × 𝐴)))
72 opelxp 5584 . . . . . . . . . . . . . . . 16 (⟨(𝐹𝑦), 𝑦⟩ ∈ (𝐵 × 𝐴) ↔ ((𝐹𝑦) ∈ 𝐵𝑦𝐴))
7371, 72syl6bb 288 . . . . . . . . . . . . . . 15 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → (𝑣 ∈ (𝐵 × 𝐴) ↔ ((𝐹𝑦) ∈ 𝐵𝑦𝐴)))
7473anbi2d 628 . . . . . . . . . . . . . 14 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → ((((𝐹𝑥) ∈ 𝐵𝑥𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ↔ (((𝐹𝑥) ∈ 𝐵𝑥𝐴) ∧ ((𝐹𝑦) ∈ 𝐵𝑦𝐴))))
75 fvex 6676 . . . . . . . . . . . . . . . . 17 (𝐹𝑦) ∈ V
76 vex 3495 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
7775, 76op1std 7688 . . . . . . . . . . . . . . . 16 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → (1st𝑣) = (𝐹𝑦))
7877breq2d 5069 . . . . . . . . . . . . . . 15 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → ((𝐹𝑥)𝑅(1st𝑣) ↔ (𝐹𝑥)𝑅(𝐹𝑦)))
7977eqeq2d 2829 . . . . . . . . . . . . . . . 16 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → ((𝐹𝑥) = (1st𝑣) ↔ (𝐹𝑥) = (𝐹𝑦)))
8075, 76op2ndd 7689 . . . . . . . . . . . . . . . . 17 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → (2nd𝑣) = 𝑦)
8180breq2d 5069 . . . . . . . . . . . . . . . 16 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → (𝑥𝑆(2nd𝑣) ↔ 𝑥𝑆𝑦))
8279, 81anbi12d 630 . . . . . . . . . . . . . . 15 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → (((𝐹𝑥) = (1st𝑣) ∧ 𝑥𝑆(2nd𝑣)) ↔ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)))
8378, 82orbi12d 912 . . . . . . . . . . . . . 14 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → (((𝐹𝑥)𝑅(1st𝑣) ∨ ((𝐹𝑥) = (1st𝑣) ∧ 𝑥𝑆(2nd𝑣))) ↔ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦))))
8474, 83anbi12d 630 . . . . . . . . . . . . 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)))))
8519, 24, 70, 84, 11brab 5421 . . . . . . . . . . . 12 (⟨(𝐹𝑥), 𝑥𝑄⟨(𝐹𝑦), 𝑦⟩ ↔ ((((𝐹𝑥) ∈ 𝐵𝑥𝐴) ∧ ((𝐹𝑦) ∈ 𝐵𝑦𝐴)) ∧ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦))))
8658, 85syl6rbbr 291 . . . . . . . . . . 11 ((𝐹:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (⟨(𝐹𝑥), 𝑥𝑄⟨(𝐹𝑦), 𝑦⟩ ↔ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦))))
8748, 50, 863bitrrd 307 . . . . . . . . . 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 5123 . . . . . . . 8 (𝐹:𝐴𝐵 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ (𝐺𝑥)𝑄(𝐺𝑦))})
9040, 89syl5eq 2865 . . . . . . 7 (𝐹:𝐴𝐵𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ (𝐺𝑥)𝑄(𝐺𝑦))})
91 isoeq3 7061 . . . . . . 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 247 . . . . 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 7072 . . . 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 6056 . . . . 5 (𝐺𝑤) = (𝐺𝑤)
98 fnwe.5 . . . . . . 7 (𝜑 → (𝐹𝑤) ∈ V)
99 vex 3495 . . . . . . 7 𝑤 ∈ V
100 xpexg 7462 . . . . . . 7 (((𝐹𝑤) ∈ V ∧ 𝑤 ∈ V) → ((𝐹𝑤) × 𝑤) ∈ V)
10198, 99, 100sylancl 586 . . . . . 6 (𝜑 → ((𝐹𝑤) × 𝑤) ∈ V)
102 imadmres 6084 . . . . . . 7 (𝐺 “ dom (𝐺𝑤)) = (𝐺𝑤)
103 dmres 5868 . . . . . . . . . . 11 dom (𝐺𝑤) = (𝑤 ∩ dom 𝐺)
104103elin2 4171 . . . . . . . . . 10 (𝑥 ∈ dom (𝐺𝑤) ↔ (𝑥𝑤𝑥 ∈ dom 𝐺))
105 simprr 769 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → 𝑥 ∈ dom 𝐺)
106 f1dm 6572 . . . . . . . . . . . . . . 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 2912 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → 𝑥𝐴)
110109, 20syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → (𝐺𝑥) = ⟨(𝐹𝑥), 𝑥⟩)
1111ffnd 6508 . . . . . . . . . . . . . . 15 (𝜑𝐹 Fn 𝐴)
112111adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → 𝐹 Fn 𝐴)
113 dmres 5868 . . . . . . . . . . . . . . 15 dom (𝐹𝑤) = (𝑤 ∩ dom 𝐹)
114 inss2 4203 . . . . . . . . . . . . . . . 16 (𝑤 ∩ dom 𝐹) ⊆ dom 𝐹
115 fndm 6448 . . . . . . . . . . . . . . . . 17 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
116112, 115syl 17 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → dom 𝐹 = 𝐴)
117114, 116sseqtrid 4016 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → (𝑤 ∩ dom 𝐹) ⊆ 𝐴)
118113, 117eqsstrid 4012 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → dom (𝐹𝑤) ⊆ 𝐴)
119 simprl 767 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → 𝑥𝑤)
120109, 116eleqtrrd 2913 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → 𝑥 ∈ dom 𝐹)
121113elin2 4171 . . . . . . . . . . . . . . 15 (𝑥 ∈ dom (𝐹𝑤) ↔ (𝑥𝑤𝑥 ∈ dom 𝐹))
122119, 120, 121sylanbrc 583 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → 𝑥 ∈ dom (𝐹𝑤))
123 fnfvima 6986 . . . . . . . . . . . . . 14 ((𝐹 Fn 𝐴 ∧ dom (𝐹𝑤) ⊆ 𝐴𝑥 ∈ dom (𝐹𝑤)) → (𝐹𝑥) ∈ (𝐹 “ dom (𝐹𝑤)))
124112, 118, 122, 123syl3anc 1363 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → (𝐹𝑥) ∈ (𝐹 “ dom (𝐹𝑤)))
125 imadmres 6084 . . . . . . . . . . . . 13 (𝐹 “ dom (𝐹𝑤)) = (𝐹𝑤)
126124, 125eleqtrdi 2920 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → (𝐹𝑥) ∈ (𝐹𝑤))
127126, 119opelxpd 5586 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → ⟨(𝐹𝑥), 𝑥⟩ ∈ ((𝐹𝑤) × 𝑤))
128110, 127eqeltrd 2910 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → (𝐺𝑥) ∈ ((𝐹𝑤) × 𝑤))
129104, 128sylan2b 593 . . . . . . . . 9 ((𝜑𝑥 ∈ dom (𝐺𝑤)) → (𝐺𝑥) ∈ ((𝐹𝑤) × 𝑤))
130129ralrimiva 3179 . . . . . . . 8 (𝜑 → ∀𝑥 ∈ dom (𝐺𝑤)(𝐺𝑥) ∈ ((𝐹𝑤) × 𝑤))
131 f1fun 6570 . . . . . . . . . 10 (𝐺:𝐴1-1→(𝐵 × 𝐴) → Fun 𝐺)
1321, 34, 1313syl 18 . . . . . . . . 9 (𝜑 → Fun 𝐺)
133 resss 5871 . . . . . . . . . 10 (𝐺𝑤) ⊆ 𝐺
134 dmss 5764 . . . . . . . . . 10 ((𝐺𝑤) ⊆ 𝐺 → dom (𝐺𝑤) ⊆ dom 𝐺)
135133, 134ax-mp 5 . . . . . . . . 9 dom (𝐺𝑤) ⊆ dom 𝐺
136 funimass4 6723 . . . . . . . . 9 ((Fun 𝐺 ∧ dom (𝐺𝑤) ⊆ dom 𝐺) → ((𝐺 “ dom (𝐺𝑤)) ⊆ ((𝐹𝑤) × 𝑤) ↔ ∀𝑥 ∈ dom (𝐺𝑤)(𝐺𝑥) ∈ ((𝐹𝑤) × 𝑤)))
137132, 135, 136sylancl 586 . . . . . . . 8 (𝜑 → ((𝐺 “ dom (𝐺𝑤)) ⊆ ((𝐹𝑤) × 𝑤) ↔ ∀𝑥 ∈ dom (𝐺𝑤)(𝐺𝑥) ∈ ((𝐹𝑤) × 𝑤)))
138130, 137mpbird 258 . . . . . . 7 (𝜑 → (𝐺 “ dom (𝐺𝑤)) ⊆ ((𝐹𝑤) × 𝑤))
139102, 138eqsstrrid 4013 . . . . . 6 (𝜑 → (𝐺𝑤) ⊆ ((𝐹𝑤) × 𝑤))
140101, 139ssexd 5219 . . . . 5 (𝜑 → (𝐺𝑤) ∈ V)
14197, 140eqeltrid 2914 . . . 4 (𝜑 → (𝐺𝑤) ∈ V)
142141alrimiv 1919 . . 3 (𝜑 → ∀𝑤(𝐺𝑤) ∈ V)
143 isowe2 7092 . . 3 ((𝐺 Isom 𝑇, 𝑄 (𝐴, ran 𝐺) ∧ ∀𝑤(𝐺𝑤) ∈ V) → (𝑄 We ran 𝐺𝑇 We 𝐴))
14496, 142, 143syl2anc 584 . 2 (𝜑 → (𝑄 We ran 𝐺𝑇 We 𝐴))
14515, 144mpd 15 1 (𝜑𝑇 We 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 841  wal 1526   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  cin 3932  wss 3933  cop 4563   class class class wbr 5057  {copab 5119  cmpt 5137   We wwe 5506   × cxp 5546  ccnv 5547  dom cdm 5548  ran crn 5549  cres 5550  cima 5551  Rel wrel 5553  Fun wfun 6342   Fn wfn 6343  wf 6344  1-1wf1 6345  1-1-ontowf1o 6347  cfv 6348   Isom wiso 6349  1st c1st 7676  2nd c2nd 7677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-1st 7678  df-2nd 7679
This theorem is referenced by:  fnwe  7815
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