Theorem fnwelem 8117

Description: Lemma for fnwe 8118. (Contributed by Mario Carneiro, 10-Mar-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)

Hypotheses

Ref	Expression
fnwe.1	⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)))}
fnwe.2	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
fnwe.3	⊢ (𝜑 → 𝑅 We 𝐵)
fnwe.4	⊢ (𝜑 → 𝑆 We 𝐴)
fnwe.5	⊢ (𝜑 → (𝐹 “ 𝑤) ∈ V)
fnwelem.6	⊢ 𝑄 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ (𝐵 × 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ∧ ((1st ‘𝑢)𝑅(1st ‘𝑣) ∨ ((1st ‘𝑢) = (1st ‘𝑣) ∧ (2nd ‘𝑢)𝑆(2nd ‘𝑣))))}
fnwelem.7	⊢ 𝐺 = (𝑧 ∈ 𝐴 ↦ ⟨(𝐹‘𝑧), 𝑧⟩)

Assertion

Ref	Expression
fnwelem	⊢ (𝜑 → 𝑇 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

Step	Hyp	Ref	Expression
1		fnwe.2	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2		ffvelcdm 7084	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝐵)
3		simpr 486	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
4	2, 3	opelxpd 5716	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑧 ∈ 𝐴) → ⟨(𝐹‘𝑧), 𝑧⟩ ∈ (𝐵 × 𝐴))
5		fnwelem.7	. . . . 5 ⊢ 𝐺 = (𝑧 ∈ 𝐴 ↦ ⟨(𝐹‘𝑧), 𝑧⟩)
6	4, 5	fmptd 7114	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐺:𝐴⟶(𝐵 × 𝐴))
7		frn 6725	. . . 4 ⊢ (𝐺:𝐴⟶(𝐵 × 𝐴) → ran 𝐺 ⊆ (𝐵 × 𝐴))
8	1, 6, 7	3syl 18	. . 3 ⊢ (𝜑 → ran 𝐺 ⊆ (𝐵 × 𝐴))
9		fnwe.3	. . . 4 ⊢ (𝜑 → 𝑅 We 𝐵)
10		fnwe.4	. . . 4 ⊢ (𝜑 → 𝑆 We 𝐴)
11		fnwelem.6	. . . . 5 ⊢ 𝑄 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ (𝐵 × 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ∧ ((1st ‘𝑢)𝑅(1st ‘𝑣) ∨ ((1st ‘𝑢) = (1st ‘𝑣) ∧ (2nd ‘𝑢)𝑆(2nd ‘𝑣))))}
12	11	wexp 8116	. . . 4 ⊢ ((𝑅 We 𝐵 ∧ 𝑆 We 𝐴) → 𝑄 We (𝐵 × 𝐴))
13	9, 10, 12	syl2anc 585	. . 3 ⊢ (𝜑 → 𝑄 We (𝐵 × 𝐴))
14		wess 5664	. . 3 ⊢ (ran 𝐺 ⊆ (𝐵 × 𝐴) → (𝑄 We (𝐵 × 𝐴) → 𝑄 We ran 𝐺))
15	8, 13, 14	sylc 65	. 2 ⊢ (𝜑 → 𝑄 We ran 𝐺)
16		fveq2 6892	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
17		id 22	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → 𝑧 = 𝑥)
18	16, 17	opeq12d 4882	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → ⟨(𝐹‘𝑧), 𝑧⟩ = ⟨(𝐹‘𝑥), 𝑥⟩)
19		opex 5465	. . . . . . . . . . 11 ⊢ ⟨(𝐹‘𝑥), 𝑥⟩ ∈ V
20	18, 5, 19	fvmpt 6999	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (𝐺‘𝑥) = ⟨(𝐹‘𝑥), 𝑥⟩)
21		fveq2 6892	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → (𝐹‘𝑧) = (𝐹‘𝑦))
22		id 22	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → 𝑧 = 𝑦)
23	21, 22	opeq12d 4882	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → ⟨(𝐹‘𝑧), 𝑧⟩ = ⟨(𝐹‘𝑦), 𝑦⟩)
24		opex 5465	. . . . . . . . . . 11 ⊢ ⟨(𝐹‘𝑦), 𝑦⟩ ∈ V
25	23, 5, 24	fvmpt 6999	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐴 → (𝐺‘𝑦) = ⟨(𝐹‘𝑦), 𝑦⟩)
26	20, 25	eqeqan12d 2747	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐺‘𝑥) = (𝐺‘𝑦) ↔ ⟨(𝐹‘𝑥), 𝑥⟩ = ⟨(𝐹‘𝑦), 𝑦⟩))
27		fvex 6905	. . . . . . . . . . 11 ⊢ (𝐹‘𝑥) ∈ V
28		vex 3479	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
29	27, 28	opth 5477	. . . . . . . . . 10 ⊢ (⟨(𝐹‘𝑥), 𝑥⟩ = ⟨(𝐹‘𝑦), 𝑦⟩ ↔ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 = 𝑦))
30	29	simprbi 498	. . . . . . . . 9 ⊢ (⟨(𝐹‘𝑥), 𝑥⟩ = ⟨(𝐹‘𝑦), 𝑦⟩ → 𝑥 = 𝑦)
31	26, 30	syl6bi 253	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐺‘𝑥) = (𝐺‘𝑦) → 𝑥 = 𝑦))
32	31	rgen2 3198	. . . . . . 7 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐺‘𝑥) = (𝐺‘𝑦) → 𝑥 = 𝑦)
33		dff13 7254	. . . . . . 7 ⊢ (𝐺:𝐴–1-1→(𝐵 × 𝐴) ↔ (𝐺:𝐴⟶(𝐵 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐺‘𝑥) = (𝐺‘𝑦) → 𝑥 = 𝑦)))
34	6, 32, 33	sylanblrc 591	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → 𝐺:𝐴–1-1→(𝐵 × 𝐴))
35		f1f1orn 6845	. . . . . 6 ⊢ (𝐺:𝐴–1-1→(𝐵 × 𝐴) → 𝐺:𝐴–1-1-onto→ran 𝐺)
36		f1ocnv 6846	. . . . . 6 ⊢ (𝐺:𝐴–1-1-onto→ran 𝐺 → ◡𝐺:ran 𝐺–1-1-onto→𝐴)
37	1, 34, 35, 36	4syl 19	. . . . 5 ⊢ (𝜑 → ◡𝐺:ran 𝐺–1-1-onto→𝐴)
38		eqid 2733	. . . . . . 7 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (◡◡𝐺‘𝑥)𝑄(◡◡𝐺‘𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (◡◡𝐺‘𝑥)𝑄(◡◡𝐺‘𝑦))}
39	38	f1oiso2 7349	. . . . . 6 ⊢ (◡𝐺:ran 𝐺–1-1-onto→𝐴 → ◡𝐺 Isom 𝑄, {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (◡◡𝐺‘𝑥)𝑄(◡◡𝐺‘𝑦))} (ran 𝐺, 𝐴))
40		fnwe.1	. . . . . . . 8 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)))}
41		frel 6723	. . . . . . . . . . . . . . . 16 ⊢ (𝐺:𝐴⟶(𝐵 × 𝐴) → Rel 𝐺)
42		dfrel2 6189	. . . . . . . . . . . . . . . 16 ⊢ (Rel 𝐺 ↔ ◡◡𝐺 = 𝐺)
43	41, 42	sylib 217	. . . . . . . . . . . . . . 15 ⊢ (𝐺:𝐴⟶(𝐵 × 𝐴) → ◡◡𝐺 = 𝐺)
44	43	fveq1d 6894	. . . . . . . . . . . . . 14 ⊢ (𝐺:𝐴⟶(𝐵 × 𝐴) → (◡◡𝐺‘𝑥) = (𝐺‘𝑥))
45	43	fveq1d 6894	. . . . . . . . . . . . . 14 ⊢ (𝐺:𝐴⟶(𝐵 × 𝐴) → (◡◡𝐺‘𝑦) = (𝐺‘𝑦))
46	44, 45	breq12d 5162	. . . . . . . . . . . . 13 ⊢ (𝐺:𝐴⟶(𝐵 × 𝐴) → ((◡◡𝐺‘𝑥)𝑄(◡◡𝐺‘𝑦) ↔ (𝐺‘𝑥)𝑄(𝐺‘𝑦)))
47	6, 46	syl 17	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶𝐵 → ((◡◡𝐺‘𝑥)𝑄(◡◡𝐺‘𝑦) ↔ (𝐺‘𝑥)𝑄(𝐺‘𝑦)))
48	47	adantr 482	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((◡◡𝐺‘𝑥)𝑄(◡◡𝐺‘𝑦) ↔ (𝐺‘𝑥)𝑄(𝐺‘𝑦)))
49	20, 25	breqan12d 5165	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐺‘𝑥)𝑄(𝐺‘𝑦) ↔ ⟨(𝐹‘𝑥), 𝑥⟩𝑄⟨(𝐹‘𝑦), 𝑦⟩))
50	49	adantl 483	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐺‘𝑥)𝑄(𝐺‘𝑦) ↔ ⟨(𝐹‘𝑥), 𝑥⟩𝑄⟨(𝐹‘𝑦), 𝑦⟩))
51		eleq1 2822	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = ⟨(𝐹‘𝑥), 𝑥⟩ → (𝑢 ∈ (𝐵 × 𝐴) ↔ ⟨(𝐹‘𝑥), 𝑥⟩ ∈ (𝐵 × 𝐴)))
52		opelxp 5713	. . . . . . . . . . . . . . . 16 ⊢ (⟨(𝐹‘𝑥), 𝑥⟩ ∈ (𝐵 × 𝐴) ↔ ((𝐹‘𝑥) ∈ 𝐵 ∧ 𝑥 ∈ 𝐴))
53	51, 52	bitrdi 287	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = ⟨(𝐹‘𝑥), 𝑥⟩ → (𝑢 ∈ (𝐵 × 𝐴) ↔ ((𝐹‘𝑥) ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)))
54	53	anbi1d 631	. . . . . . . . . . . . . 14 ⊢ (𝑢 = ⟨(𝐹‘𝑥), 𝑥⟩ → ((𝑢 ∈ (𝐵 × 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ↔ (((𝐹‘𝑥) ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴))))
55	27, 28	op1std 7985	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = ⟨(𝐹‘𝑥), 𝑥⟩ → (1st ‘𝑢) = (𝐹‘𝑥))
56	55	breq1d 5159	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = ⟨(𝐹‘𝑥), 𝑥⟩ → ((1st ‘𝑢)𝑅(1st ‘𝑣) ↔ (𝐹‘𝑥)𝑅(1st ‘𝑣)))
57	55	eqeq1d 2735	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = ⟨(𝐹‘𝑥), 𝑥⟩ → ((1st ‘𝑢) = (1st ‘𝑣) ↔ (𝐹‘𝑥) = (1st ‘𝑣)))
58	27, 28	op2ndd 7986	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = ⟨(𝐹‘𝑥), 𝑥⟩ → (2nd ‘𝑢) = 𝑥)
59	58	breq1d 5159	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = ⟨(𝐹‘𝑥), 𝑥⟩ → ((2nd ‘𝑢)𝑆(2nd ‘𝑣) ↔ 𝑥𝑆(2nd ‘𝑣)))
60	57, 59	anbi12d 632	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = ⟨(𝐹‘𝑥), 𝑥⟩ → (((1st ‘𝑢) = (1st ‘𝑣) ∧ (2nd ‘𝑢)𝑆(2nd ‘𝑣)) ↔ ((𝐹‘𝑥) = (1st ‘𝑣) ∧ 𝑥𝑆(2nd ‘𝑣))))
61	56, 60	orbi12d 918	. . . . . . . . . . . . . 14 ⊢ (𝑢 = ⟨(𝐹‘𝑥), 𝑥⟩ → (((1st ‘𝑢)𝑅(1st ‘𝑣) ∨ ((1st ‘𝑢) = (1st ‘𝑣) ∧ (2nd ‘𝑢)𝑆(2nd ‘𝑣))) ↔ ((𝐹‘𝑥)𝑅(1st ‘𝑣) ∨ ((𝐹‘𝑥) = (1st ‘𝑣) ∧ 𝑥𝑆(2nd ‘𝑣)))))
62	54, 61	anbi12d 632	. . . . . . . . . . . . 13 ⊢ (𝑢 = ⟨(𝐹‘𝑥), 𝑥⟩ → (((𝑢 ∈ (𝐵 × 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ∧ ((1st ‘𝑢)𝑅(1st ‘𝑣) ∨ ((1st ‘𝑢) = (1st ‘𝑣) ∧ (2nd ‘𝑢)𝑆(2nd ‘𝑣)))) ↔ ((((𝐹‘𝑥) ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ∧ ((𝐹‘𝑥)𝑅(1st ‘𝑣) ∨ ((𝐹‘𝑥) = (1st ‘𝑣) ∧ 𝑥𝑆(2nd ‘𝑣))))))
63		eleq1 2822	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = ⟨(𝐹‘𝑦), 𝑦⟩ → (𝑣 ∈ (𝐵 × 𝐴) ↔ ⟨(𝐹‘𝑦), 𝑦⟩ ∈ (𝐵 × 𝐴)))
64		opelxp 5713	. . . . . . . . . . . . . . . 16 ⊢ (⟨(𝐹‘𝑦), 𝑦⟩ ∈ (𝐵 × 𝐴) ↔ ((𝐹‘𝑦) ∈ 𝐵 ∧ 𝑦 ∈ 𝐴))
65	63, 64	bitrdi 287	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = ⟨(𝐹‘𝑦), 𝑦⟩ → (𝑣 ∈ (𝐵 × 𝐴) ↔ ((𝐹‘𝑦) ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)))
66	65	anbi2d 630	. . . . . . . . . . . . . 14 ⊢ (𝑣 = ⟨(𝐹‘𝑦), 𝑦⟩ → ((((𝐹‘𝑥) ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ↔ (((𝐹‘𝑥) ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ ((𝐹‘𝑦) ∈ 𝐵 ∧ 𝑦 ∈ 𝐴))))
67		fvex 6905	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹‘𝑦) ∈ V
68		vex 3479	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
69	67, 68	op1std 7985	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = ⟨(𝐹‘𝑦), 𝑦⟩ → (1st ‘𝑣) = (𝐹‘𝑦))
70	69	breq2d 5161	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = ⟨(𝐹‘𝑦), 𝑦⟩ → ((𝐹‘𝑥)𝑅(1st ‘𝑣) ↔ (𝐹‘𝑥)𝑅(𝐹‘𝑦)))
71	69	eqeq2d 2744	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = ⟨(𝐹‘𝑦), 𝑦⟩ → ((𝐹‘𝑥) = (1st ‘𝑣) ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))
72	67, 68	op2ndd 7986	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = ⟨(𝐹‘𝑦), 𝑦⟩ → (2nd ‘𝑣) = 𝑦)
73	72	breq2d 5161	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = ⟨(𝐹‘𝑦), 𝑦⟩ → (𝑥𝑆(2nd ‘𝑣) ↔ 𝑥𝑆𝑦))
74	71, 73	anbi12d 632	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = ⟨(𝐹‘𝑦), 𝑦⟩ → (((𝐹‘𝑥) = (1st ‘𝑣) ∧ 𝑥𝑆(2nd ‘𝑣)) ↔ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)))
75	70, 74	orbi12d 918	. . . . . . . . . . . . . 14 ⊢ (𝑣 = ⟨(𝐹‘𝑦), 𝑦⟩ → (((𝐹‘𝑥)𝑅(1st ‘𝑣) ∨ ((𝐹‘𝑥) = (1st ‘𝑣) ∧ 𝑥𝑆(2nd ‘𝑣))) ↔ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦))))
76	66, 75	anbi12d 632	. . . . . . . . . . . . 13 ⊢ (𝑣 = ⟨(𝐹‘𝑦), 𝑦⟩ → (((((𝐹‘𝑥) ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ∧ ((𝐹‘𝑥)𝑅(1st ‘𝑣) ∨ ((𝐹‘𝑥) = (1st ‘𝑣) ∧ 𝑥𝑆(2nd ‘𝑣)))) ↔ ((((𝐹‘𝑥) ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ ((𝐹‘𝑦) ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)))))
77	19, 24, 62, 76, 11	brab 5544	. . . . . . . . . . . 12 ⊢ (⟨(𝐹‘𝑥), 𝑥⟩𝑄⟨(𝐹‘𝑦), 𝑦⟩ ↔ ((((𝐹‘𝑥) ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ ((𝐹‘𝑦) ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦))))
78		ffvelcdm 7084	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
79		simpr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
80	78, 79	jca 513	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) ∈ 𝐵 ∧ 𝑥 ∈ 𝐴))
81		ffvelcdm 7084	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ 𝐵)
82		simpr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
83	81, 82	jca 513	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑦) ∈ 𝐵 ∧ 𝑦 ∈ 𝐴))
84	80, 83	anim12dan 620	. . . . . . . . . . . . 13 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((𝐹‘𝑥) ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ ((𝐹‘𝑦) ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)))
85	84	biantrurd 534	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)) ↔ ((((𝐹‘𝑥) ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ ((𝐹‘𝑦) ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) ∧ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)))))
86	77, 85	bitr4id 290	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (⟨(𝐹‘𝑥), 𝑥⟩𝑄⟨(𝐹‘𝑦), 𝑦⟩ ↔ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦))))
87	48, 50, 86	3bitrrd 306	. . . . . . . . . 10 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)) ↔ (◡◡𝐺‘𝑥)𝑄(◡◡𝐺‘𝑦)))
88	87	pm5.32da 580	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶𝐵 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦))) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (◡◡𝐺‘𝑥)𝑄(◡◡𝐺‘𝑦))))
89	88	opabbidv 5215	. . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (◡◡𝐺‘𝑥)𝑄(◡◡𝐺‘𝑦))})
90	40, 89	eqtrid 2785	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (◡◡𝐺‘𝑥)𝑄(◡◡𝐺‘𝑦))})
91		isoeq3 7316	. . . . . . 7 ⊢ (𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (◡◡𝐺‘𝑥)𝑄(◡◡𝐺‘𝑦))} → (◡𝐺 Isom 𝑄, 𝑇 (ran 𝐺, 𝐴) ↔ ◡𝐺 Isom 𝑄, {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (◡◡𝐺‘𝑥)𝑄(◡◡𝐺‘𝑦))} (ran 𝐺, 𝐴)))
92	90, 91	syl 17	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (◡𝐺 Isom 𝑄, 𝑇 (ran 𝐺, 𝐴) ↔ ◡𝐺 Isom 𝑄, {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (◡◡𝐺‘𝑥)𝑄(◡◡𝐺‘𝑦))} (ran 𝐺, 𝐴)))
93	39, 92	imbitrrid 245	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → (◡𝐺:ran 𝐺–1-1-onto→𝐴 → ◡𝐺 Isom 𝑄, 𝑇 (ran 𝐺, 𝐴)))
94	1, 37, 93	sylc 65	. . . 4 ⊢ (𝜑 → ◡𝐺 Isom 𝑄, 𝑇 (ran 𝐺, 𝐴))
95		isocnv 7327	. . . 4 ⊢ (◡𝐺 Isom 𝑄, 𝑇 (ran 𝐺, 𝐴) → ◡◡𝐺 Isom 𝑇, 𝑄 (𝐴, ran 𝐺))
96	94, 95	syl 17	. . 3 ⊢ (𝜑 → ◡◡𝐺 Isom 𝑇, 𝑄 (𝐴, ran 𝐺))
97		imacnvcnv 6206	. . . . 5 ⊢ (◡◡𝐺 “ 𝑤) = (𝐺 “ 𝑤)
98		fnwe.5	. . . . . . 7 ⊢ (𝜑 → (𝐹 “ 𝑤) ∈ V)
99		vex 3479	. . . . . . 7 ⊢ 𝑤 ∈ V
100		xpexg 7737	. . . . . . 7 ⊢ (((𝐹 “ 𝑤) ∈ V ∧ 𝑤 ∈ V) → ((𝐹 “ 𝑤) × 𝑤) ∈ V)
101	98, 99, 100	sylancl 587	. . . . . 6 ⊢ (𝜑 → ((𝐹 “ 𝑤) × 𝑤) ∈ V)
102		imadmres 6234	. . . . . . 7 ⊢ (𝐺 “ dom (𝐺 ↾ 𝑤)) = (𝐺 “ 𝑤)
103		dmres 6004	. . . . . . . . . . 11 ⊢ dom (𝐺 ↾ 𝑤) = (𝑤 ∩ dom 𝐺)
104	103	elin2 4198	. . . . . . . . . 10 ⊢ (𝑥 ∈ dom (𝐺 ↾ 𝑤) ↔ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺))
105		simprr 772	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺)) → 𝑥 ∈ dom 𝐺)
106		f1dm 6792	. . . . . . . . . . . . . . 15 ⊢ (𝐺:𝐴–1-1→(𝐵 × 𝐴) → dom 𝐺 = 𝐴)
107	1, 34, 106	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝐺 = 𝐴)
108	107	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺)) → dom 𝐺 = 𝐴)
109	105, 108	eleqtrd 2836	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺)) → 𝑥 ∈ 𝐴)
110	109, 20	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺)) → (𝐺‘𝑥) = ⟨(𝐹‘𝑥), 𝑥⟩)
111	1	ffnd 6719	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 Fn 𝐴)
112	111	adantr 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺)) → 𝐹 Fn 𝐴)
113		dmres 6004	. . . . . . . . . . . . . . 15 ⊢ dom (𝐹 ↾ 𝑤) = (𝑤 ∩ dom 𝐹)
114		inss2 4230	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∩ dom 𝐹) ⊆ dom 𝐹
115	112	fndmd 6655	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺)) → dom 𝐹 = 𝐴)
116	114, 115	sseqtrid 4035	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺)) → (𝑤 ∩ dom 𝐹) ⊆ 𝐴)
117	113, 116	eqsstrid 4031	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺)) → dom (𝐹 ↾ 𝑤) ⊆ 𝐴)
118		simprl 770	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺)) → 𝑥 ∈ 𝑤)
119	109, 115	eleqtrrd 2837	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺)) → 𝑥 ∈ dom 𝐹)
120	113	elin2 4198	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ dom (𝐹 ↾ 𝑤) ↔ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐹))
121	118, 119, 120	sylanbrc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺)) → 𝑥 ∈ dom (𝐹 ↾ 𝑤))
122		fnfvima 7235	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn 𝐴 ∧ dom (𝐹 ↾ 𝑤) ⊆ 𝐴 ∧ 𝑥 ∈ dom (𝐹 ↾ 𝑤)) → (𝐹‘𝑥) ∈ (𝐹 “ dom (𝐹 ↾ 𝑤)))
123	112, 117, 121, 122	syl3anc 1372	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺)) → (𝐹‘𝑥) ∈ (𝐹 “ dom (𝐹 ↾ 𝑤)))
124		imadmres 6234	. . . . . . . . . . . . 13 ⊢ (𝐹 “ dom (𝐹 ↾ 𝑤)) = (𝐹 “ 𝑤)
125	123, 124	eleqtrdi 2844	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺)) → (𝐹‘𝑥) ∈ (𝐹 “ 𝑤))
126	125, 118	opelxpd 5716	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺)) → ⟨(𝐹‘𝑥), 𝑥⟩ ∈ ((𝐹 “ 𝑤) × 𝑤))
127	110, 126	eqeltrd 2834	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑤 ∧ 𝑥 ∈ dom 𝐺)) → (𝐺‘𝑥) ∈ ((𝐹 “ 𝑤) × 𝑤))
128	104, 127	sylan2b 595	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐺 ↾ 𝑤)) → (𝐺‘𝑥) ∈ ((𝐹 “ 𝑤) × 𝑤))
129	128	ralrimiva 3147	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ dom (𝐺 ↾ 𝑤)(𝐺‘𝑥) ∈ ((𝐹 “ 𝑤) × 𝑤))
130		f1fun 6790	. . . . . . . . . 10 ⊢ (𝐺:𝐴–1-1→(𝐵 × 𝐴) → Fun 𝐺)
131	1, 34, 130	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → Fun 𝐺)
132		resss 6007	. . . . . . . . . 10 ⊢ (𝐺 ↾ 𝑤) ⊆ 𝐺
133		dmss 5903	. . . . . . . . . 10 ⊢ ((𝐺 ↾ 𝑤) ⊆ 𝐺 → dom (𝐺 ↾ 𝑤) ⊆ dom 𝐺)
134	132, 133	ax-mp 5	. . . . . . . . 9 ⊢ dom (𝐺 ↾ 𝑤) ⊆ dom 𝐺
135		funimass4 6957	. . . . . . . . 9 ⊢ ((Fun 𝐺 ∧ dom (𝐺 ↾ 𝑤) ⊆ dom 𝐺) → ((𝐺 “ dom (𝐺 ↾ 𝑤)) ⊆ ((𝐹 “ 𝑤) × 𝑤) ↔ ∀𝑥 ∈ dom (𝐺 ↾ 𝑤)(𝐺‘𝑥) ∈ ((𝐹 “ 𝑤) × 𝑤)))
136	131, 134, 135	sylancl 587	. . . . . . . 8 ⊢ (𝜑 → ((𝐺 “ dom (𝐺 ↾ 𝑤)) ⊆ ((𝐹 “ 𝑤) × 𝑤) ↔ ∀𝑥 ∈ dom (𝐺 ↾ 𝑤)(𝐺‘𝑥) ∈ ((𝐹 “ 𝑤) × 𝑤)))
137	129, 136	mpbird 257	. . . . . . 7 ⊢ (𝜑 → (𝐺 “ dom (𝐺 ↾ 𝑤)) ⊆ ((𝐹 “ 𝑤) × 𝑤))
138	102, 137	eqsstrrid 4032	. . . . . 6 ⊢ (𝜑 → (𝐺 “ 𝑤) ⊆ ((𝐹 “ 𝑤) × 𝑤))
139	101, 138	ssexd 5325	. . . . 5 ⊢ (𝜑 → (𝐺 “ 𝑤) ∈ V)
140	97, 139	eqeltrid 2838	. . . 4 ⊢ (𝜑 → (◡◡𝐺 “ 𝑤) ∈ V)
141	140	alrimiv 1931	. . 3 ⊢ (𝜑 → ∀𝑤(◡◡𝐺 “ 𝑤) ∈ V)
142		isowe2 7347	. . 3 ⊢ ((◡◡𝐺 Isom 𝑇, 𝑄 (𝐴, ran 𝐺) ∧ ∀𝑤(◡◡𝐺 “ 𝑤) ∈ V) → (𝑄 We ran 𝐺 → 𝑇 We 𝐴))
143	96, 141, 142	syl2anc 585	. 2 ⊢ (𝜑 → (𝑄 We ran 𝐺 → 𝑇 We 𝐴))
144	15, 143	mpd 15	1 ⊢ (𝜑 → 𝑇 We 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846  ∀wal 1540   = wceq 1542   ∈ wcel 2107  ∀wral 3062  Vcvv 3475   ∩ cin 3948   ⊆ wss 3949  ⟨cop 4635   class class class wbr 5149  {copab 5211   ↦ cmpt 5232   We wwe 5631   × cxp 5675  ^◡ccnv 5676  dom cdm 5677  ran crn 5678   ↾ cres 5679   “ cima 5680  Rel wrel 5682  Fun wfun 6538   Fn wfn 6539  ⟶wf 6540  –1-1→wf1 6541  –1-1-onto→wf1o 6543  ‘cfv 6544   Isom wiso 6545  1^st c1st 7973  2^nd c2nd 7974

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-1st 7975  df-2nd 7976

This theorem is referenced by:  fnwe  8118