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Theorem fnwelem 7827
 Description: Lemma for fnwe 7828. (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 6851 . . . . . 6 ((𝐹:𝐴𝐵𝑧𝐴) → (𝐹𝑧) ∈ 𝐵)
3 simpr 487 . . . . . 6 ((𝐹:𝐴𝐵𝑧𝐴) → 𝑧𝐴)
42, 3opelxpd 5595 . . . . 5 ((𝐹:𝐴𝐵𝑧𝐴) → ⟨(𝐹𝑧), 𝑧⟩ ∈ (𝐵 × 𝐴))
5 fnwelem.7 . . . . 5 𝐺 = (𝑧𝐴 ↦ ⟨(𝐹𝑧), 𝑧⟩)
64, 5fmptd 6880 . . . 4 (𝐹:𝐴𝐵𝐺:𝐴⟶(𝐵 × 𝐴))
7 frn 6522 . . . 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 7826 . . . 4 ((𝑅 We 𝐵𝑆 We 𝐴) → 𝑄 We (𝐵 × 𝐴))
139, 10, 12syl2anc 586 . . 3 (𝜑𝑄 We (𝐵 × 𝐴))
14 wess 5544 . . 3 (ran 𝐺 ⊆ (𝐵 × 𝐴) → (𝑄 We (𝐵 × 𝐴) → 𝑄 We ran 𝐺))
158, 13, 14sylc 65 . 2 (𝜑𝑄 We ran 𝐺)
16 fveq2 6672 . . . . . . . . . . . 12 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
17 id 22 . . . . . . . . . . . 12 (𝑧 = 𝑥𝑧 = 𝑥)
1816, 17opeq12d 4813 . . . . . . . . . . 11 (𝑧 = 𝑥 → ⟨(𝐹𝑧), 𝑧⟩ = ⟨(𝐹𝑥), 𝑥⟩)
19 opex 5358 . . . . . . . . . . 11 ⟨(𝐹𝑥), 𝑥⟩ ∈ V
2018, 5, 19fvmpt 6770 . . . . . . . . . 10 (𝑥𝐴 → (𝐺𝑥) = ⟨(𝐹𝑥), 𝑥⟩)
21 fveq2 6672 . . . . . . . . . . . 12 (𝑧 = 𝑦 → (𝐹𝑧) = (𝐹𝑦))
22 id 22 . . . . . . . . . . . 12 (𝑧 = 𝑦𝑧 = 𝑦)
2321, 22opeq12d 4813 . . . . . . . . . . 11 (𝑧 = 𝑦 → ⟨(𝐹𝑧), 𝑧⟩ = ⟨(𝐹𝑦), 𝑦⟩)
24 opex 5358 . . . . . . . . . . 11 ⟨(𝐹𝑦), 𝑦⟩ ∈ V
2523, 5, 24fvmpt 6770 . . . . . . . . . 10 (𝑦𝐴 → (𝐺𝑦) = ⟨(𝐹𝑦), 𝑦⟩)
2620, 25eqeqan12d 2840 . . . . . . . . 9 ((𝑥𝐴𝑦𝐴) → ((𝐺𝑥) = (𝐺𝑦) ↔ ⟨(𝐹𝑥), 𝑥⟩ = ⟨(𝐹𝑦), 𝑦⟩))
27 fvex 6685 . . . . . . . . . . 11 (𝐹𝑥) ∈ V
28 vex 3499 . . . . . . . . . . 11 𝑥 ∈ V
2927, 28opth 5370 . . . . . . . . . 10 (⟨(𝐹𝑥), 𝑥⟩ = ⟨(𝐹𝑦), 𝑦⟩ ↔ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 = 𝑦))
3029simprbi 499 . . . . . . . . 9 (⟨(𝐹𝑥), 𝑥⟩ = ⟨(𝐹𝑦), 𝑦⟩ → 𝑥 = 𝑦)
3126, 30syl6bi 255 . . . . . . . 8 ((𝑥𝐴𝑦𝐴) → ((𝐺𝑥) = (𝐺𝑦) → 𝑥 = 𝑦))
3231rgen2 3205 . . . . . . 7 𝑥𝐴𝑦𝐴 ((𝐺𝑥) = (𝐺𝑦) → 𝑥 = 𝑦)
33 dff13 7015 . . . . . . 7 (𝐺:𝐴1-1→(𝐵 × 𝐴) ↔ (𝐺:𝐴⟶(𝐵 × 𝐴) ∧ ∀𝑥𝐴𝑦𝐴 ((𝐺𝑥) = (𝐺𝑦) → 𝑥 = 𝑦)))
346, 32, 33sylanblrc 592 . . . . . 6 (𝐹:𝐴𝐵𝐺:𝐴1-1→(𝐵 × 𝐴))
35 f1f1orn 6628 . . . . . 6 (𝐺:𝐴1-1→(𝐵 × 𝐴) → 𝐺:𝐴1-1-onto→ran 𝐺)
36 f1ocnv 6629 . . . . . 6 (𝐺:𝐴1-1-onto→ran 𝐺𝐺:ran 𝐺1-1-onto𝐴)
371, 34, 35, 364syl 19 . . . . 5 (𝜑𝐺:ran 𝐺1-1-onto𝐴)
38 eqid 2823 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ (𝐺𝑥)𝑄(𝐺𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ (𝐺𝑥)𝑄(𝐺𝑦))}
3938f1oiso2 7107 . . . . . 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 6521 . . . . . . . . . . . . . . . 16 (𝐺:𝐴⟶(𝐵 × 𝐴) → Rel 𝐺)
42 dfrel2 6048 . . . . . . . . . . . . . . . 16 (Rel 𝐺𝐺 = 𝐺)
4341, 42sylib 220 . . . . . . . . . . . . . . 15 (𝐺:𝐴⟶(𝐵 × 𝐴) → 𝐺 = 𝐺)
4443fveq1d 6674 . . . . . . . . . . . . . 14 (𝐺:𝐴⟶(𝐵 × 𝐴) → (𝐺𝑥) = (𝐺𝑥))
4543fveq1d 6674 . . . . . . . . . . . . . 14 (𝐺:𝐴⟶(𝐵 × 𝐴) → (𝐺𝑦) = (𝐺𝑦))
4644, 45breq12d 5081 . . . . . . . . . . . . 13 (𝐺:𝐴⟶(𝐵 × 𝐴) → ((𝐺𝑥)𝑄(𝐺𝑦) ↔ (𝐺𝑥)𝑄(𝐺𝑦)))
476, 46syl 17 . . . . . . . . . . . 12 (𝐹:𝐴𝐵 → ((𝐺𝑥)𝑄(𝐺𝑦) ↔ (𝐺𝑥)𝑄(𝐺𝑦)))
4847adantr 483 . . . . . . . . . . 11 ((𝐹:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐺𝑥)𝑄(𝐺𝑦) ↔ (𝐺𝑥)𝑄(𝐺𝑦)))
4920, 25breqan12d 5084 . . . . . . . . . . . 12 ((𝑥𝐴𝑦𝐴) → ((𝐺𝑥)𝑄(𝐺𝑦) ↔ ⟨(𝐹𝑥), 𝑥𝑄⟨(𝐹𝑦), 𝑦⟩))
5049adantl 484 . . . . . . . . . . 11 ((𝐹:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐺𝑥)𝑄(𝐺𝑦) ↔ ⟨(𝐹𝑥), 𝑥𝑄⟨(𝐹𝑦), 𝑦⟩))
51 ffvelrn 6851 . . . . . . . . . . . . . . 15 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
52 simpr 487 . . . . . . . . . . . . . . 15 ((𝐹:𝐴𝐵𝑥𝐴) → 𝑥𝐴)
5351, 52jca 514 . . . . . . . . . . . . . 14 ((𝐹:𝐴𝐵𝑥𝐴) → ((𝐹𝑥) ∈ 𝐵𝑥𝐴))
54 ffvelrn 6851 . . . . . . . . . . . . . . 15 ((𝐹:𝐴𝐵𝑦𝐴) → (𝐹𝑦) ∈ 𝐵)
55 simpr 487 . . . . . . . . . . . . . . 15 ((𝐹:𝐴𝐵𝑦𝐴) → 𝑦𝐴)
5654, 55jca 514 . . . . . . . . . . . . . 14 ((𝐹:𝐴𝐵𝑦𝐴) → ((𝐹𝑦) ∈ 𝐵𝑦𝐴))
5753, 56anim12dan 620 . . . . . . . . . . . . 13 ((𝐹:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (((𝐹𝑥) ∈ 𝐵𝑥𝐴) ∧ ((𝐹𝑦) ∈ 𝐵𝑦𝐴)))
5857biantrurd 535 . . . . . . . . . . . 12 ((𝐹:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)) ↔ ((((𝐹𝑥) ∈ 𝐵𝑥𝐴) ∧ ((𝐹𝑦) ∈ 𝐵𝑦𝐴)) ∧ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)))))
59 eleq1 2902 . . . . . . . . . . . . . . . 16 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → (𝑢 ∈ (𝐵 × 𝐴) ↔ ⟨(𝐹𝑥), 𝑥⟩ ∈ (𝐵 × 𝐴)))
60 opelxp 5593 . . . . . . . . . . . . . . . 16 (⟨(𝐹𝑥), 𝑥⟩ ∈ (𝐵 × 𝐴) ↔ ((𝐹𝑥) ∈ 𝐵𝑥𝐴))
6159, 60syl6bb 289 . . . . . . . . . . . . . . 15 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → (𝑢 ∈ (𝐵 × 𝐴) ↔ ((𝐹𝑥) ∈ 𝐵𝑥𝐴)))
6261anbi1d 631 . . . . . . . . . . . . . 14 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → ((𝑢 ∈ (𝐵 × 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ↔ (((𝐹𝑥) ∈ 𝐵𝑥𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴))))
6327, 28op1std 7701 . . . . . . . . . . . . . . . 16 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → (1st𝑢) = (𝐹𝑥))
6463breq1d 5078 . . . . . . . . . . . . . . 15 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → ((1st𝑢)𝑅(1st𝑣) ↔ (𝐹𝑥)𝑅(1st𝑣)))
6563eqeq1d 2825 . . . . . . . . . . . . . . . 16 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → ((1st𝑢) = (1st𝑣) ↔ (𝐹𝑥) = (1st𝑣)))
6627, 28op2ndd 7702 . . . . . . . . . . . . . . . . 17 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → (2nd𝑢) = 𝑥)
6766breq1d 5078 . . . . . . . . . . . . . . . 16 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → ((2nd𝑢)𝑆(2nd𝑣) ↔ 𝑥𝑆(2nd𝑣)))
6865, 67anbi12d 632 . . . . . . . . . . . . . . 15 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → (((1st𝑢) = (1st𝑣) ∧ (2nd𝑢)𝑆(2nd𝑣)) ↔ ((𝐹𝑥) = (1st𝑣) ∧ 𝑥𝑆(2nd𝑣))))
6964, 68orbi12d 915 . . . . . . . . . . . . . 14 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → (((1st𝑢)𝑅(1st𝑣) ∨ ((1st𝑢) = (1st𝑣) ∧ (2nd𝑢)𝑆(2nd𝑣))) ↔ ((𝐹𝑥)𝑅(1st𝑣) ∨ ((𝐹𝑥) = (1st𝑣) ∧ 𝑥𝑆(2nd𝑣)))))
7062, 69anbi12d 632 . . . . . . . . . . . . 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 2902 . . . . . . . . . . . . . . . 16 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → (𝑣 ∈ (𝐵 × 𝐴) ↔ ⟨(𝐹𝑦), 𝑦⟩ ∈ (𝐵 × 𝐴)))
72 opelxp 5593 . . . . . . . . . . . . . . . 16 (⟨(𝐹𝑦), 𝑦⟩ ∈ (𝐵 × 𝐴) ↔ ((𝐹𝑦) ∈ 𝐵𝑦𝐴))
7371, 72syl6bb 289 . . . . . . . . . . . . . . 15 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → (𝑣 ∈ (𝐵 × 𝐴) ↔ ((𝐹𝑦) ∈ 𝐵𝑦𝐴)))
7473anbi2d 630 . . . . . . . . . . . . . 14 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → ((((𝐹𝑥) ∈ 𝐵𝑥𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ↔ (((𝐹𝑥) ∈ 𝐵𝑥𝐴) ∧ ((𝐹𝑦) ∈ 𝐵𝑦𝐴))))
75 fvex 6685 . . . . . . . . . . . . . . . . 17 (𝐹𝑦) ∈ V
76 vex 3499 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
7775, 76op1std 7701 . . . . . . . . . . . . . . . 16 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → (1st𝑣) = (𝐹𝑦))
7877breq2d 5080 . . . . . . . . . . . . . . 15 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → ((𝐹𝑥)𝑅(1st𝑣) ↔ (𝐹𝑥)𝑅(𝐹𝑦)))
7977eqeq2d 2834 . . . . . . . . . . . . . . . 16 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → ((𝐹𝑥) = (1st𝑣) ↔ (𝐹𝑥) = (𝐹𝑦)))
8075, 76op2ndd 7702 . . . . . . . . . . . . . . . . 17 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → (2nd𝑣) = 𝑦)
8180breq2d 5080 . . . . . . . . . . . . . . . 16 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → (𝑥𝑆(2nd𝑣) ↔ 𝑥𝑆𝑦))
8279, 81anbi12d 632 . . . . . . . . . . . . . . 15 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → (((𝐹𝑥) = (1st𝑣) ∧ 𝑥𝑆(2nd𝑣)) ↔ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)))
8378, 82orbi12d 915 . . . . . . . . . . . . . 14 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → (((𝐹𝑥)𝑅(1st𝑣) ∨ ((𝐹𝑥) = (1st𝑣) ∧ 𝑥𝑆(2nd𝑣))) ↔ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦))))
8474, 83anbi12d 632 . . . . . . . . . . . . 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 5432 . . . . . . . . . . . 12 (⟨(𝐹𝑥), 𝑥𝑄⟨(𝐹𝑦), 𝑦⟩ ↔ ((((𝐹𝑥) ∈ 𝐵𝑥𝐴) ∧ ((𝐹𝑦) ∈ 𝐵𝑦𝐴)) ∧ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦))))
8658, 85syl6rbbr 292 . . . . . . . . . . 11 ((𝐹:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (⟨(𝐹𝑥), 𝑥𝑄⟨(𝐹𝑦), 𝑦⟩ ↔ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦))))
8748, 50, 863bitrrd 308 . . . . . . . . . 10 ((𝐹:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)) ↔ (𝐺𝑥)𝑄(𝐺𝑦)))
8887pm5.32da 581 . . . . . . . . 9 (𝐹:𝐴𝐵 → (((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦))) ↔ ((𝑥𝐴𝑦𝐴) ∧ (𝐺𝑥)𝑄(𝐺𝑦))))
8988opabbidv 5134 . . . . . . . 8 (𝐹:𝐴𝐵 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ (𝐺𝑥)𝑄(𝐺𝑦))})
9040, 89syl5eq 2870 . . . . . . 7 (𝐹:𝐴𝐵𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ (𝐺𝑥)𝑄(𝐺𝑦))})
91 isoeq3 7074 . . . . . . 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 248 . . . . 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 7085 . . . 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 6065 . . . . 5 (𝐺𝑤) = (𝐺𝑤)
98 fnwe.5 . . . . . . 7 (𝜑 → (𝐹𝑤) ∈ V)
99 vex 3499 . . . . . . 7 𝑤 ∈ V
100 xpexg 7475 . . . . . . 7 (((𝐹𝑤) ∈ V ∧ 𝑤 ∈ V) → ((𝐹𝑤) × 𝑤) ∈ V)
10198, 99, 100sylancl 588 . . . . . 6 (𝜑 → ((𝐹𝑤) × 𝑤) ∈ V)
102 imadmres 6093 . . . . . . 7 (𝐺 “ dom (𝐺𝑤)) = (𝐺𝑤)
103 dmres 5877 . . . . . . . . . . 11 dom (𝐺𝑤) = (𝑤 ∩ dom 𝐺)
104103elin2 4176 . . . . . . . . . 10 (𝑥 ∈ dom (𝐺𝑤) ↔ (𝑥𝑤𝑥 ∈ dom 𝐺))
105 simprr 771 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → 𝑥 ∈ dom 𝐺)
106 f1dm 6581 . . . . . . . . . . . . . . 15 (𝐺:𝐴1-1→(𝐵 × 𝐴) → dom 𝐺 = 𝐴)
1071, 34, 1063syl 18 . . . . . . . . . . . . . 14 (𝜑 → dom 𝐺 = 𝐴)
108107adantr 483 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → dom 𝐺 = 𝐴)
109105, 108eleqtrd 2917 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → 𝑥𝐴)
110109, 20syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → (𝐺𝑥) = ⟨(𝐹𝑥), 𝑥⟩)
1111ffnd 6517 . . . . . . . . . . . . . . 15 (𝜑𝐹 Fn 𝐴)
112111adantr 483 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → 𝐹 Fn 𝐴)
113 dmres 5877 . . . . . . . . . . . . . . 15 dom (𝐹𝑤) = (𝑤 ∩ dom 𝐹)
114 inss2 4208 . . . . . . . . . . . . . . . 16 (𝑤 ∩ dom 𝐹) ⊆ dom 𝐹
115 fndm 6457 . . . . . . . . . . . . . . . . 17 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
116112, 115syl 17 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → dom 𝐹 = 𝐴)
117114, 116sseqtrid 4021 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → (𝑤 ∩ dom 𝐹) ⊆ 𝐴)
118113, 117eqsstrid 4017 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → dom (𝐹𝑤) ⊆ 𝐴)
119 simprl 769 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → 𝑥𝑤)
120109, 116eleqtrrd 2918 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → 𝑥 ∈ dom 𝐹)
121113elin2 4176 . . . . . . . . . . . . . . 15 (𝑥 ∈ dom (𝐹𝑤) ↔ (𝑥𝑤𝑥 ∈ dom 𝐹))
122119, 120, 121sylanbrc 585 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → 𝑥 ∈ dom (𝐹𝑤))
123 fnfvima 6997 . . . . . . . . . . . . . 14 ((𝐹 Fn 𝐴 ∧ dom (𝐹𝑤) ⊆ 𝐴𝑥 ∈ dom (𝐹𝑤)) → (𝐹𝑥) ∈ (𝐹 “ dom (𝐹𝑤)))
124112, 118, 122, 123syl3anc 1367 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → (𝐹𝑥) ∈ (𝐹 “ dom (𝐹𝑤)))
125 imadmres 6093 . . . . . . . . . . . . 13 (𝐹 “ dom (𝐹𝑤)) = (𝐹𝑤)
126124, 125eleqtrdi 2925 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → (𝐹𝑥) ∈ (𝐹𝑤))
127126, 119opelxpd 5595 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → ⟨(𝐹𝑥), 𝑥⟩ ∈ ((𝐹𝑤) × 𝑤))
128110, 127eqeltrd 2915 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → (𝐺𝑥) ∈ ((𝐹𝑤) × 𝑤))
129104, 128sylan2b 595 . . . . . . . . 9 ((𝜑𝑥 ∈ dom (𝐺𝑤)) → (𝐺𝑥) ∈ ((𝐹𝑤) × 𝑤))
130129ralrimiva 3184 . . . . . . . 8 (𝜑 → ∀𝑥 ∈ dom (𝐺𝑤)(𝐺𝑥) ∈ ((𝐹𝑤) × 𝑤))
131 f1fun 6579 . . . . . . . . . 10 (𝐺:𝐴1-1→(𝐵 × 𝐴) → Fun 𝐺)
1321, 34, 1313syl 18 . . . . . . . . 9 (𝜑 → Fun 𝐺)
133 resss 5880 . . . . . . . . . 10 (𝐺𝑤) ⊆ 𝐺
134 dmss 5773 . . . . . . . . . 10 ((𝐺𝑤) ⊆ 𝐺 → dom (𝐺𝑤) ⊆ dom 𝐺)
135133, 134ax-mp 5 . . . . . . . . 9 dom (𝐺𝑤) ⊆ dom 𝐺
136 funimass4 6732 . . . . . . . . 9 ((Fun 𝐺 ∧ dom (𝐺𝑤) ⊆ dom 𝐺) → ((𝐺 “ dom (𝐺𝑤)) ⊆ ((𝐹𝑤) × 𝑤) ↔ ∀𝑥 ∈ dom (𝐺𝑤)(𝐺𝑥) ∈ ((𝐹𝑤) × 𝑤)))
137132, 135, 136sylancl 588 . . . . . . . 8 (𝜑 → ((𝐺 “ dom (𝐺𝑤)) ⊆ ((𝐹𝑤) × 𝑤) ↔ ∀𝑥 ∈ dom (𝐺𝑤)(𝐺𝑥) ∈ ((𝐹𝑤) × 𝑤)))
138130, 137mpbird 259 . . . . . . 7 (𝜑 → (𝐺 “ dom (𝐺𝑤)) ⊆ ((𝐹𝑤) × 𝑤))
139102, 138eqsstrrid 4018 . . . . . 6 (𝜑 → (𝐺𝑤) ⊆ ((𝐹𝑤) × 𝑤))
140101, 139ssexd 5230 . . . . 5 (𝜑 → (𝐺𝑤) ∈ V)
14197, 140eqeltrid 2919 . . . 4 (𝜑 → (𝐺𝑤) ∈ V)
142141alrimiv 1928 . . 3 (𝜑 → ∀𝑤(𝐺𝑤) ∈ V)
143 isowe2 7105 . . 3 ((𝐺 Isom 𝑇, 𝑄 (𝐴, ran 𝐺) ∧ ∀𝑤(𝐺𝑤) ∈ V) → (𝑄 We ran 𝐺𝑇 We 𝐴))
14496, 142, 143syl2anc 586 . 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 208   ∧ wa 398   ∨ wo 843  ∀wal 1535   = wceq 1537   ∈ wcel 2114  ∀wral 3140  Vcvv 3496   ∩ cin 3937   ⊆ wss 3938  ⟨cop 4575   class class class wbr 5068  {copab 5130   ↦ cmpt 5148   We wwe 5515   × cxp 5555  ◡ccnv 5556  dom cdm 5557  ran crn 5558   ↾ cres 5559   “ cima 5560  Rel wrel 5562  Fun wfun 6351   Fn wfn 6352  ⟶wf 6353  –1-1→wf1 6354  –1-1-onto→wf1o 6356  ‘cfv 6357   Isom wiso 6358  1st c1st 7689  2nd c2nd 7690 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-int 4879  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-isom 6366  df-1st 7691  df-2nd 7692 This theorem is referenced by:  fnwe  7828
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