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Theorem fnwelem 7337
Description: Lemma for fnwe 7338. (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 6397 . . . . . 6 ((𝐹:𝐴𝐵𝑧𝐴) → (𝐹𝑧) ∈ 𝐵)
3 simpr 476 . . . . . 6 ((𝐹:𝐴𝐵𝑧𝐴) → 𝑧𝐴)
42, 3opelxpd 5183 . . . . 5 ((𝐹:𝐴𝐵𝑧𝐴) → ⟨(𝐹𝑧), 𝑧⟩ ∈ (𝐵 × 𝐴))
5 fnwelem.7 . . . . 5 𝐺 = (𝑧𝐴 ↦ ⟨(𝐹𝑧), 𝑧⟩)
64, 5fmptd 6425 . . . 4 (𝐹:𝐴𝐵𝐺:𝐴⟶(𝐵 × 𝐴))
7 frn 6091 . . . 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 7336 . . . 4 ((𝑅 We 𝐵𝑆 We 𝐴) → 𝑄 We (𝐵 × 𝐴))
139, 10, 12syl2anc 694 . . 3 (𝜑𝑄 We (𝐵 × 𝐴))
14 wess 5130 . . 3 (ran 𝐺 ⊆ (𝐵 × 𝐴) → (𝑄 We (𝐵 × 𝐴) → 𝑄 We ran 𝐺))
158, 13, 14sylc 65 . 2 (𝜑𝑄 We ran 𝐺)
16 fveq2 6229 . . . . . . . . . . . 12 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
17 id 22 . . . . . . . . . . . 12 (𝑧 = 𝑥𝑧 = 𝑥)
1816, 17opeq12d 4441 . . . . . . . . . . 11 (𝑧 = 𝑥 → ⟨(𝐹𝑧), 𝑧⟩ = ⟨(𝐹𝑥), 𝑥⟩)
19 opex 4962 . . . . . . . . . . 11 ⟨(𝐹𝑥), 𝑥⟩ ∈ V
2018, 5, 19fvmpt 6321 . . . . . . . . . 10 (𝑥𝐴 → (𝐺𝑥) = ⟨(𝐹𝑥), 𝑥⟩)
21 fveq2 6229 . . . . . . . . . . . 12 (𝑧 = 𝑦 → (𝐹𝑧) = (𝐹𝑦))
22 id 22 . . . . . . . . . . . 12 (𝑧 = 𝑦𝑧 = 𝑦)
2321, 22opeq12d 4441 . . . . . . . . . . 11 (𝑧 = 𝑦 → ⟨(𝐹𝑧), 𝑧⟩ = ⟨(𝐹𝑦), 𝑦⟩)
24 opex 4962 . . . . . . . . . . 11 ⟨(𝐹𝑦), 𝑦⟩ ∈ V
2523, 5, 24fvmpt 6321 . . . . . . . . . 10 (𝑦𝐴 → (𝐺𝑦) = ⟨(𝐹𝑦), 𝑦⟩)
2620, 25eqeqan12d 2667 . . . . . . . . 9 ((𝑥𝐴𝑦𝐴) → ((𝐺𝑥) = (𝐺𝑦) ↔ ⟨(𝐹𝑥), 𝑥⟩ = ⟨(𝐹𝑦), 𝑦⟩))
27 fvex 6239 . . . . . . . . . . 11 (𝐹𝑥) ∈ V
28 vex 3234 . . . . . . . . . . 11 𝑥 ∈ V
2927, 28opth 4974 . . . . . . . . . 10 (⟨(𝐹𝑥), 𝑥⟩ = ⟨(𝐹𝑦), 𝑦⟩ ↔ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 = 𝑦))
3029simprbi 479 . . . . . . . . 9 (⟨(𝐹𝑥), 𝑥⟩ = ⟨(𝐹𝑦), 𝑦⟩ → 𝑥 = 𝑦)
3126, 30syl6bi 243 . . . . . . . 8 ((𝑥𝐴𝑦𝐴) → ((𝐺𝑥) = (𝐺𝑦) → 𝑥 = 𝑦))
3231rgen2a 3006 . . . . . . 7 𝑥𝐴𝑦𝐴 ((𝐺𝑥) = (𝐺𝑦) → 𝑥 = 𝑦)
33 dff13 6552 . . . . . . 7 (𝐺:𝐴1-1→(𝐵 × 𝐴) ↔ (𝐺:𝐴⟶(𝐵 × 𝐴) ∧ ∀𝑥𝐴𝑦𝐴 ((𝐺𝑥) = (𝐺𝑦) → 𝑥 = 𝑦)))
346, 32, 33sylanblrc 698 . . . . . 6 (𝐹:𝐴𝐵𝐺:𝐴1-1→(𝐵 × 𝐴))
35 f1f1orn 6186 . . . . . 6 (𝐺:𝐴1-1→(𝐵 × 𝐴) → 𝐺:𝐴1-1-onto→ran 𝐺)
36 f1ocnv 6187 . . . . . 6 (𝐺:𝐴1-1-onto→ran 𝐺𝐺:ran 𝐺1-1-onto𝐴)
371, 34, 35, 364syl 19 . . . . 5 (𝜑𝐺:ran 𝐺1-1-onto𝐴)
38 eqid 2651 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ (𝐺𝑥)𝑄(𝐺𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ (𝐺𝑥)𝑄(𝐺𝑦))}
3938f1oiso2 6642 . . . . . 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 6088 . . . . . . . . . . . . . . . 16 (𝐺:𝐴⟶(𝐵 × 𝐴) → Rel 𝐺)
42 dfrel2 5618 . . . . . . . . . . . . . . . 16 (Rel 𝐺𝐺 = 𝐺)
4341, 42sylib 208 . . . . . . . . . . . . . . 15 (𝐺:𝐴⟶(𝐵 × 𝐴) → 𝐺 = 𝐺)
4443fveq1d 6231 . . . . . . . . . . . . . 14 (𝐺:𝐴⟶(𝐵 × 𝐴) → (𝐺𝑥) = (𝐺𝑥))
4543fveq1d 6231 . . . . . . . . . . . . . 14 (𝐺:𝐴⟶(𝐵 × 𝐴) → (𝐺𝑦) = (𝐺𝑦))
4644, 45breq12d 4698 . . . . . . . . . . . . 13 (𝐺:𝐴⟶(𝐵 × 𝐴) → ((𝐺𝑥)𝑄(𝐺𝑦) ↔ (𝐺𝑥)𝑄(𝐺𝑦)))
476, 46syl 17 . . . . . . . . . . . 12 (𝐹:𝐴𝐵 → ((𝐺𝑥)𝑄(𝐺𝑦) ↔ (𝐺𝑥)𝑄(𝐺𝑦)))
4847adantr 480 . . . . . . . . . . 11 ((𝐹:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐺𝑥)𝑄(𝐺𝑦) ↔ (𝐺𝑥)𝑄(𝐺𝑦)))
4920, 25breqan12d 4701 . . . . . . . . . . . 12 ((𝑥𝐴𝑦𝐴) → ((𝐺𝑥)𝑄(𝐺𝑦) ↔ ⟨(𝐹𝑥), 𝑥𝑄⟨(𝐹𝑦), 𝑦⟩))
5049adantl 481 . . . . . . . . . . 11 ((𝐹:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → ((𝐺𝑥)𝑄(𝐺𝑦) ↔ ⟨(𝐹𝑥), 𝑥𝑄⟨(𝐹𝑦), 𝑦⟩))
51 ffvelrn 6397 . . . . . . . . . . . . . . 15 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
52 simpr 476 . . . . . . . . . . . . . . 15 ((𝐹:𝐴𝐵𝑥𝐴) → 𝑥𝐴)
5351, 52jca 553 . . . . . . . . . . . . . 14 ((𝐹:𝐴𝐵𝑥𝐴) → ((𝐹𝑥) ∈ 𝐵𝑥𝐴))
54 ffvelrn 6397 . . . . . . . . . . . . . . 15 ((𝐹:𝐴𝐵𝑦𝐴) → (𝐹𝑦) ∈ 𝐵)
55 simpr 476 . . . . . . . . . . . . . . 15 ((𝐹:𝐴𝐵𝑦𝐴) → 𝑦𝐴)
5654, 55jca 553 . . . . . . . . . . . . . 14 ((𝐹:𝐴𝐵𝑦𝐴) → ((𝐹𝑦) ∈ 𝐵𝑦𝐴))
5753, 56anim12dan 900 . . . . . . . . . . . . 13 ((𝐹:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (((𝐹𝑥) ∈ 𝐵𝑥𝐴) ∧ ((𝐹𝑦) ∈ 𝐵𝑦𝐴)))
5857biantrurd 528 . . . . . . . . . . . 12 ((𝐹:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)) ↔ ((((𝐹𝑥) ∈ 𝐵𝑥𝐴) ∧ ((𝐹𝑦) ∈ 𝐵𝑦𝐴)) ∧ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)))))
59 eleq1 2718 . . . . . . . . . . . . . . . 16 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → (𝑢 ∈ (𝐵 × 𝐴) ↔ ⟨(𝐹𝑥), 𝑥⟩ ∈ (𝐵 × 𝐴)))
60 opelxp 5180 . . . . . . . . . . . . . . . 16 (⟨(𝐹𝑥), 𝑥⟩ ∈ (𝐵 × 𝐴) ↔ ((𝐹𝑥) ∈ 𝐵𝑥𝐴))
6159, 60syl6bb 276 . . . . . . . . . . . . . . 15 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → (𝑢 ∈ (𝐵 × 𝐴) ↔ ((𝐹𝑥) ∈ 𝐵𝑥𝐴)))
6261anbi1d 741 . . . . . . . . . . . . . 14 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → ((𝑢 ∈ (𝐵 × 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ↔ (((𝐹𝑥) ∈ 𝐵𝑥𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴))))
6327, 28op1std 7220 . . . . . . . . . . . . . . . 16 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → (1st𝑢) = (𝐹𝑥))
6463breq1d 4695 . . . . . . . . . . . . . . 15 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → ((1st𝑢)𝑅(1st𝑣) ↔ (𝐹𝑥)𝑅(1st𝑣)))
6563eqeq1d 2653 . . . . . . . . . . . . . . . 16 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → ((1st𝑢) = (1st𝑣) ↔ (𝐹𝑥) = (1st𝑣)))
6627, 28op2ndd 7221 . . . . . . . . . . . . . . . . 17 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → (2nd𝑢) = 𝑥)
6766breq1d 4695 . . . . . . . . . . . . . . . 16 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → ((2nd𝑢)𝑆(2nd𝑣) ↔ 𝑥𝑆(2nd𝑣)))
6865, 67anbi12d 747 . . . . . . . . . . . . . . 15 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → (((1st𝑢) = (1st𝑣) ∧ (2nd𝑢)𝑆(2nd𝑣)) ↔ ((𝐹𝑥) = (1st𝑣) ∧ 𝑥𝑆(2nd𝑣))))
6964, 68orbi12d 746 . . . . . . . . . . . . . 14 (𝑢 = ⟨(𝐹𝑥), 𝑥⟩ → (((1st𝑢)𝑅(1st𝑣) ∨ ((1st𝑢) = (1st𝑣) ∧ (2nd𝑢)𝑆(2nd𝑣))) ↔ ((𝐹𝑥)𝑅(1st𝑣) ∨ ((𝐹𝑥) = (1st𝑣) ∧ 𝑥𝑆(2nd𝑣)))))
7062, 69anbi12d 747 . . . . . . . . . . . . 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 2718 . . . . . . . . . . . . . . . 16 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → (𝑣 ∈ (𝐵 × 𝐴) ↔ ⟨(𝐹𝑦), 𝑦⟩ ∈ (𝐵 × 𝐴)))
72 opelxp 5180 . . . . . . . . . . . . . . . 16 (⟨(𝐹𝑦), 𝑦⟩ ∈ (𝐵 × 𝐴) ↔ ((𝐹𝑦) ∈ 𝐵𝑦𝐴))
7371, 72syl6bb 276 . . . . . . . . . . . . . . 15 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → (𝑣 ∈ (𝐵 × 𝐴) ↔ ((𝐹𝑦) ∈ 𝐵𝑦𝐴)))
7473anbi2d 740 . . . . . . . . . . . . . 14 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → ((((𝐹𝑥) ∈ 𝐵𝑥𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ↔ (((𝐹𝑥) ∈ 𝐵𝑥𝐴) ∧ ((𝐹𝑦) ∈ 𝐵𝑦𝐴))))
75 fvex 6239 . . . . . . . . . . . . . . . . 17 (𝐹𝑦) ∈ V
76 vex 3234 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
7775, 76op1std 7220 . . . . . . . . . . . . . . . 16 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → (1st𝑣) = (𝐹𝑦))
7877breq2d 4697 . . . . . . . . . . . . . . 15 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → ((𝐹𝑥)𝑅(1st𝑣) ↔ (𝐹𝑥)𝑅(𝐹𝑦)))
7977eqeq2d 2661 . . . . . . . . . . . . . . . 16 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → ((𝐹𝑥) = (1st𝑣) ↔ (𝐹𝑥) = (𝐹𝑦)))
8075, 76op2ndd 7221 . . . . . . . . . . . . . . . . 17 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → (2nd𝑣) = 𝑦)
8180breq2d 4697 . . . . . . . . . . . . . . . 16 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → (𝑥𝑆(2nd𝑣) ↔ 𝑥𝑆𝑦))
8279, 81anbi12d 747 . . . . . . . . . . . . . . 15 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → (((𝐹𝑥) = (1st𝑣) ∧ 𝑥𝑆(2nd𝑣)) ↔ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)))
8378, 82orbi12d 746 . . . . . . . . . . . . . 14 (𝑣 = ⟨(𝐹𝑦), 𝑦⟩ → (((𝐹𝑥)𝑅(1st𝑣) ∨ ((𝐹𝑥) = (1st𝑣) ∧ 𝑥𝑆(2nd𝑣))) ↔ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦))))
8474, 83anbi12d 747 . . . . . . . . . . . . 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 5027 . . . . . . . . . . . 12 (⟨(𝐹𝑥), 𝑥𝑄⟨(𝐹𝑦), 𝑦⟩ ↔ ((((𝐹𝑥) ∈ 𝐵𝑥𝐴) ∧ ((𝐹𝑦) ∈ 𝐵𝑦𝐴)) ∧ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦))))
8658, 85syl6rbbr 279 . . . . . . . . . . 11 ((𝐹:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (⟨(𝐹𝑥), 𝑥𝑄⟨(𝐹𝑦), 𝑦⟩ ↔ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦))))
8748, 50, 863bitrrd 295 . . . . . . . . . 10 ((𝐹:𝐴𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)) ↔ (𝐺𝑥)𝑄(𝐺𝑦)))
8887pm5.32da 674 . . . . . . . . 9 (𝐹:𝐴𝐵 → (((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦))) ↔ ((𝑥𝐴𝑦𝐴) ∧ (𝐺𝑥)𝑄(𝐺𝑦))))
8988opabbidv 4749 . . . . . . . 8 (𝐹:𝐴𝐵 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ (𝐺𝑥)𝑄(𝐺𝑦))})
9040, 89syl5eq 2697 . . . . . . 7 (𝐹:𝐴𝐵𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ (𝐺𝑥)𝑄(𝐺𝑦))})
91 isoeq3 6609 . . . . . . 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 236 . . . . 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 6620 . . . 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 5634 . . . . 5 (𝐺𝑤) = (𝐺𝑤)
98 fnwe.5 . . . . . . 7 (𝜑 → (𝐹𝑤) ∈ V)
99 vex 3234 . . . . . . 7 𝑤 ∈ V
100 xpexg 7002 . . . . . . 7 (((𝐹𝑤) ∈ V ∧ 𝑤 ∈ V) → ((𝐹𝑤) × 𝑤) ∈ V)
10198, 99, 100sylancl 695 . . . . . 6 (𝜑 → ((𝐹𝑤) × 𝑤) ∈ V)
102 imadmres 5665 . . . . . . 7 (𝐺 “ dom (𝐺𝑤)) = (𝐺𝑤)
103 dmres 5454 . . . . . . . . . . 11 dom (𝐺𝑤) = (𝑤 ∩ dom 𝐺)
104103elin2 3834 . . . . . . . . . 10 (𝑥 ∈ dom (𝐺𝑤) ↔ (𝑥𝑤𝑥 ∈ dom 𝐺))
105 simprr 811 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → 𝑥 ∈ dom 𝐺)
106 f1dm 6143 . . . . . . . . . . . . . . 15 (𝐺:𝐴1-1→(𝐵 × 𝐴) → dom 𝐺 = 𝐴)
1071, 34, 1063syl 18 . . . . . . . . . . . . . 14 (𝜑 → dom 𝐺 = 𝐴)
108107adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → dom 𝐺 = 𝐴)
109105, 108eleqtrd 2732 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → 𝑥𝐴)
110109, 20syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → (𝐺𝑥) = ⟨(𝐹𝑥), 𝑥⟩)
1111ffnd 6084 . . . . . . . . . . . . . . 15 (𝜑𝐹 Fn 𝐴)
112111adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → 𝐹 Fn 𝐴)
113 dmres 5454 . . . . . . . . . . . . . . 15 dom (𝐹𝑤) = (𝑤 ∩ dom 𝐹)
114 inss2 3867 . . . . . . . . . . . . . . . 16 (𝑤 ∩ dom 𝐹) ⊆ dom 𝐹
115 fndm 6028 . . . . . . . . . . . . . . . . 17 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
116112, 115syl 17 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → dom 𝐹 = 𝐴)
117114, 116syl5sseq 3686 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → (𝑤 ∩ dom 𝐹) ⊆ 𝐴)
118113, 117syl5eqss 3682 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → dom (𝐹𝑤) ⊆ 𝐴)
119 simprl 809 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → 𝑥𝑤)
120109, 116eleqtrrd 2733 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → 𝑥 ∈ dom 𝐹)
121113elin2 3834 . . . . . . . . . . . . . . 15 (𝑥 ∈ dom (𝐹𝑤) ↔ (𝑥𝑤𝑥 ∈ dom 𝐹))
122119, 120, 121sylanbrc 699 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → 𝑥 ∈ dom (𝐹𝑤))
123 fnfvima 6536 . . . . . . . . . . . . . 14 ((𝐹 Fn 𝐴 ∧ dom (𝐹𝑤) ⊆ 𝐴𝑥 ∈ dom (𝐹𝑤)) → (𝐹𝑥) ∈ (𝐹 “ dom (𝐹𝑤)))
124112, 118, 122, 123syl3anc 1366 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → (𝐹𝑥) ∈ (𝐹 “ dom (𝐹𝑤)))
125 imadmres 5665 . . . . . . . . . . . . 13 (𝐹 “ dom (𝐹𝑤)) = (𝐹𝑤)
126124, 125syl6eleq 2740 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → (𝐹𝑥) ∈ (𝐹𝑤))
127126, 119opelxpd 5183 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → ⟨(𝐹𝑥), 𝑥⟩ ∈ ((𝐹𝑤) × 𝑤))
128110, 127eqeltrd 2730 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑤𝑥 ∈ dom 𝐺)) → (𝐺𝑥) ∈ ((𝐹𝑤) × 𝑤))
129104, 128sylan2b 491 . . . . . . . . 9 ((𝜑𝑥 ∈ dom (𝐺𝑤)) → (𝐺𝑥) ∈ ((𝐹𝑤) × 𝑤))
130129ralrimiva 2995 . . . . . . . 8 (𝜑 → ∀𝑥 ∈ dom (𝐺𝑤)(𝐺𝑥) ∈ ((𝐹𝑤) × 𝑤))
131 f1fun 6141 . . . . . . . . . 10 (𝐺:𝐴1-1→(𝐵 × 𝐴) → Fun 𝐺)
1321, 34, 1313syl 18 . . . . . . . . 9 (𝜑 → Fun 𝐺)
133 resss 5457 . . . . . . . . . 10 (𝐺𝑤) ⊆ 𝐺
134 dmss 5355 . . . . . . . . . 10 ((𝐺𝑤) ⊆ 𝐺 → dom (𝐺𝑤) ⊆ dom 𝐺)
135133, 134ax-mp 5 . . . . . . . . 9 dom (𝐺𝑤) ⊆ dom 𝐺
136 funimass4 6286 . . . . . . . . 9 ((Fun 𝐺 ∧ dom (𝐺𝑤) ⊆ dom 𝐺) → ((𝐺 “ dom (𝐺𝑤)) ⊆ ((𝐹𝑤) × 𝑤) ↔ ∀𝑥 ∈ dom (𝐺𝑤)(𝐺𝑥) ∈ ((𝐹𝑤) × 𝑤)))
137132, 135, 136sylancl 695 . . . . . . . 8 (𝜑 → ((𝐺 “ dom (𝐺𝑤)) ⊆ ((𝐹𝑤) × 𝑤) ↔ ∀𝑥 ∈ dom (𝐺𝑤)(𝐺𝑥) ∈ ((𝐹𝑤) × 𝑤)))
138130, 137mpbird 247 . . . . . . 7 (𝜑 → (𝐺 “ dom (𝐺𝑤)) ⊆ ((𝐹𝑤) × 𝑤))
139102, 138syl5eqssr 3683 . . . . . 6 (𝜑 → (𝐺𝑤) ⊆ ((𝐹𝑤) × 𝑤))
140101, 139ssexd 4838 . . . . 5 (𝜑 → (𝐺𝑤) ∈ V)
14197, 140syl5eqel 2734 . . . 4 (𝜑 → (𝐺𝑤) ∈ V)
142141alrimiv 1895 . . 3 (𝜑 → ∀𝑤(𝐺𝑤) ∈ V)
143 isowe2 6640 . . 3 ((𝐺 Isom 𝑇, 𝑄 (𝐴, ran 𝐺) ∧ ∀𝑤(𝐺𝑤) ∈ V) → (𝑄 We ran 𝐺𝑇 We 𝐴))
14496, 142, 143syl2anc 694 . 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 196  wo 382  wa 383  wal 1521   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  cin 3606  wss 3607  cop 4216   class class class wbr 4685  {copab 4745  cmpt 4762   We wwe 5101   × cxp 5141  ccnv 5142  dom cdm 5143  ran crn 5144  cres 5145  cima 5146  Rel wrel 5148  Fun wfun 5920   Fn wfn 5921  wf 5922  1-1wf1 5923  1-1-ontowf1o 5925  cfv 5926   Isom wiso 5927  1st c1st 7208  2nd c2nd 7209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-1st 7210  df-2nd 7211
This theorem is referenced by:  fnwe  7338
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