Theorem fmptcof2 28839

Description: Composition of two functions expressed as ordered-pair class abstractions. (Contributed by FL, 21-Jun-2012.) (Revised by Mario Carneiro, 24-Jul-2014.) (Revised by Thierry Arnoux, 10-May-2017.)

Hypotheses

Ref	Expression
fmptcof2.x	⊢ Ⅎ𝑥𝑆
fmptcof2.y	⊢ Ⅎ𝑦𝑇
fmptcof2.1	⊢ Ⅎ𝑥𝐴
fmptcof2.2	⊢ Ⅎ𝑥𝐵
fmptcof2.3	⊢ Ⅎ𝑥𝜑
fmptcof2.4	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑅 ∈ 𝐵)
fmptcof2.5	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑅))
fmptcof2.6	⊢ (𝜑 → 𝐺 = (𝑦 ∈ 𝐵 ↦ 𝑆))
fmptcof2.7	⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇)

Assertion

Ref	Expression
fmptcof2	⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝑇))

Distinct variable groups:   𝑥,𝑦   𝑦,𝐵   𝑦,𝑅

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥)   𝑅(𝑥)   𝑆(𝑥,𝑦)   𝑇(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem fmptcof2

Dummy variables 𝑣 𝑢 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relco 5550	. 2 ⊢ Rel (𝐺 ∘ 𝐹)
2		funmpt 5840	. . 3 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝑇)
3		funrel 5821	. . 3 ⊢ (Fun (𝑥 ∈ 𝐴 ↦ 𝑇) → Rel (𝑥 ∈ 𝐴 ↦ 𝑇))
4	2, 3	ax-mp 5	. 2 ⊢ Rel (𝑥 ∈ 𝐴 ↦ 𝑇)
5		fmptcof2.3	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝜑
6		fmptcof2.1	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝐴
7		fmptcof2.2	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝐵
8		fmptcof2.4	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑅 ∈ 𝐵)
9	8	r19.21bi 2916	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ 𝐵)
10		eqid 2610	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑅) = (𝑥 ∈ 𝐴 ↦ 𝑅)
11	5, 6, 7, 9, 10	fmptdF 28836	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑅):𝐴⟶𝐵)
12		fmptcof2.5	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑅))
13	12	feq1d 5943	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ (𝑥 ∈ 𝐴 ↦ 𝑅):𝐴⟶𝐵))
14	11, 13	mpbird 246	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
15		ffun 5961	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
16	14, 15	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → Fun 𝐹)
17		funbrfv 6144	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → (𝑧𝐹𝑢 → (𝐹‘𝑧) = 𝑢))
18	17	imp 444	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑧𝐹𝑢) → (𝐹‘𝑧) = 𝑢)
19	16, 18	sylan 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧𝐹𝑢) → (𝐹‘𝑧) = 𝑢)
20	19	eqcomd 2616	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧𝐹𝑢) → 𝑢 = (𝐹‘𝑧))
21	20	a1d 25	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧𝐹𝑢) → (𝑢𝐺𝑤 → 𝑢 = (𝐹‘𝑧)))
22	21	expimpd 627	. . . . . 6 ⊢ (𝜑 → ((𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤) → 𝑢 = (𝐹‘𝑧)))
23	22	pm4.71rd 665	. . . . 5 ⊢ (𝜑 → ((𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤) ↔ (𝑢 = (𝐹‘𝑧) ∧ (𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤))))
24	23	exbidv 1837	. . . 4 ⊢ (𝜑 → (∃𝑢(𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤) ↔ ∃𝑢(𝑢 = (𝐹‘𝑧) ∧ (𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤))))
25		fvex 6113	. . . . . 6 ⊢ (𝐹‘𝑧) ∈ V
26		breq2 4587	. . . . . . 7 ⊢ (𝑢 = (𝐹‘𝑧) → (𝑧𝐹𝑢 ↔ 𝑧𝐹(𝐹‘𝑧)))
27		breq1 4586	. . . . . . 7 ⊢ (𝑢 = (𝐹‘𝑧) → (𝑢𝐺𝑤 ↔ (𝐹‘𝑧)𝐺𝑤))
28	26, 27	anbi12d 743	. . . . . 6 ⊢ (𝑢 = (𝐹‘𝑧) → ((𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤) ↔ (𝑧𝐹(𝐹‘𝑧) ∧ (𝐹‘𝑧)𝐺𝑤)))
29	25, 28	ceqsexv 3215	. . . . 5 ⊢ (∃𝑢(𝑢 = (𝐹‘𝑧) ∧ (𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤)) ↔ (𝑧𝐹(𝐹‘𝑧) ∧ (𝐹‘𝑧)𝐺𝑤))
30		funfvbrb 6238	. . . . . . . . 9 ⊢ (Fun 𝐹 → (𝑧 ∈ dom 𝐹 ↔ 𝑧𝐹(𝐹‘𝑧)))
31	16, 30	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑧 ∈ dom 𝐹 ↔ 𝑧𝐹(𝐹‘𝑧)))
32		fdm 5964	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
33	14, 32	syl 17	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 = 𝐴)
34	33	eleq2d 2673	. . . . . . . 8 ⊢ (𝜑 → (𝑧 ∈ dom 𝐹 ↔ 𝑧 ∈ 𝐴))
35	31, 34	bitr3d 269	. . . . . . 7 ⊢ (𝜑 → (𝑧𝐹(𝐹‘𝑧) ↔ 𝑧 ∈ 𝐴))
36	12	fveq1d 6105	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑧) = ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧))
37		fmptcof2.6	. . . . . . . 8 ⊢ (𝜑 → 𝐺 = (𝑦 ∈ 𝐵 ↦ 𝑆))
38		eqidd 2611	. . . . . . . 8 ⊢ (𝜑 → 𝑤 = 𝑤)
39	36, 37, 38	breq123d 4597	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝑧)𝐺𝑤 ↔ ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤))
40	35, 39	anbi12d 743	. . . . . 6 ⊢ (𝜑 → ((𝑧𝐹(𝐹‘𝑧) ∧ (𝐹‘𝑧)𝐺𝑤) ↔ (𝑧 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤)))
41	6	nfcri 2745	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
42		nffvmpt1 6111	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)
43		fmptcof2.x	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥𝑆
44	7, 43	nfmpt 4674	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐵 ↦ 𝑆)
45		nfcv 2751	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝑤
46	42, 44, 45	nfbr 4629	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤
47		nfcsb1v 3515	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝑇
48	47	nfeq2 2766	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 𝑤 = ⦋𝑧 / 𝑥⦌𝑇
49	46, 48	nfbi 1821	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)
50	5, 49	nfim 1813	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝜑 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))
51	41, 50	nfim 1813	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 → (𝜑 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))
52		eleq1 2676	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
53		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧))
54	53	breq1d 4593	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤))
55		csbeq1a 3508	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → 𝑇 = ⦋𝑧 / 𝑥⦌𝑇)
56	55	eqeq2d 2620	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑤 = 𝑇 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))
57	54, 56	bibi12d 334	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = 𝑇) ↔ (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))
58	57	imbi2d 329	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ((𝜑 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = 𝑇)) ↔ (𝜑 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))))
59	52, 58	imbi12d 333	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 → (𝜑 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = 𝑇))) ↔ (𝑧 ∈ 𝐴 → (𝜑 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))))
60		vex 3176	. . . . . . . . . . . 12 ⊢ 𝑤 ∈ V
61		nfv 1830	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦 𝑅 ∈ 𝐵
62		nfcv 2751	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑦𝑤
63		fmptcof2.y	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑦𝑇
64	62, 63	nfeq 2762	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦 𝑤 = 𝑇
65	61, 64	nfan 1816	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦(𝑅 ∈ 𝐵 ∧ 𝑤 = 𝑇)
66		simpl 472	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 = 𝑅 ∧ 𝑢 = 𝑤) → 𝑦 = 𝑅)
67	66	eleq1d 2672	. . . . . . . . . . . . . 14 ⊢ ((𝑦 = 𝑅 ∧ 𝑢 = 𝑤) → (𝑦 ∈ 𝐵 ↔ 𝑅 ∈ 𝐵))
68		simpr 476	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 = 𝑅 ∧ 𝑢 = 𝑤) → 𝑢 = 𝑤)
69		fmptcof2.7	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇)
70	69	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 = 𝑅 ∧ 𝑢 = 𝑤) → 𝑆 = 𝑇)
71	68, 70	eqeq12d 2625	. . . . . . . . . . . . . 14 ⊢ ((𝑦 = 𝑅 ∧ 𝑢 = 𝑤) → (𝑢 = 𝑆 ↔ 𝑤 = 𝑇))
72	67, 71	anbi12d 743	. . . . . . . . . . . . 13 ⊢ ((𝑦 = 𝑅 ∧ 𝑢 = 𝑤) → ((𝑦 ∈ 𝐵 ∧ 𝑢 = 𝑆) ↔ (𝑅 ∈ 𝐵 ∧ 𝑤 = 𝑇)))
73		df-mpt 4645	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐵 ↦ 𝑆) = {⟨𝑦, 𝑢⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑢 = 𝑆)}
74	65, 72, 73	brabgaf 28800	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ 𝐵 ∧ 𝑤 ∈ V) → (𝑅(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ (𝑅 ∈ 𝐵 ∧ 𝑤 = 𝑇)))
75	9, 60, 74	sylancl 693	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ (𝑅 ∈ 𝐵 ∧ 𝑤 = 𝑇)))
76		simpr 476	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
77	6	fvmpt2f 6192	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑅 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥) = 𝑅)
78	76, 9, 77	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥) = 𝑅)
79	78	breq1d 4593	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑅(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤))
80	9	biantrurd 528	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑤 = 𝑇 ↔ (𝑅 ∈ 𝐵 ∧ 𝑤 = 𝑇)))
81	75, 79, 80	3bitr4d 299	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = 𝑇))
82	81	expcom 450	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = 𝑇)))
83	51, 59, 82	chvar 2250	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 → (𝜑 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))
84	83	impcom 445	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))
85	84	pm5.32da 671	. . . . . 6 ⊢ (𝜑 → ((𝑧 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))
86	40, 85	bitrd 267	. . . . 5 ⊢ (𝜑 → ((𝑧𝐹(𝐹‘𝑧) ∧ (𝐹‘𝑧)𝐺𝑤) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))
87	29, 86	syl5bb 271	. . . 4 ⊢ (𝜑 → (∃𝑢(𝑢 = (𝐹‘𝑧) ∧ (𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤)) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))
88	24, 87	bitrd 267	. . 3 ⊢ (𝜑 → (∃𝑢(𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))
89		vex 3176	. . . 4 ⊢ 𝑧 ∈ V
90	89, 60	opelco 5215	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝐺 ∘ 𝐹) ↔ ∃𝑢(𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤))
91		df-mpt 4645	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑇) = {⟨𝑥, 𝑣⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑣 = 𝑇)}
92	91	eleq2i 2680	. . . 4 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝑥 ∈ 𝐴 ↦ 𝑇) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑣⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑣 = 𝑇)})
93	47	nfeq2 2766	. . . . . 6 ⊢ Ⅎ𝑥 𝑣 = ⦋𝑧 / 𝑥⦌𝑇
94	41, 93	nfan 1816	. . . . 5 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝑇)
95		nfv 1830	. . . . 5 ⊢ Ⅎ𝑣(𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)
96	55	eqeq2d 2620	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑣 = 𝑇 ↔ 𝑣 = ⦋𝑧 / 𝑥⦌𝑇))
97	52, 96	anbi12d 743	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝑣 = 𝑇) ↔ (𝑧 ∈ 𝐴 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝑇)))
98		eqeq1 2614	. . . . . 6 ⊢ (𝑣 = 𝑤 → (𝑣 = ⦋𝑧 / 𝑥⦌𝑇 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))
99	98	anbi2d 736	. . . . 5 ⊢ (𝑣 = 𝑤 → ((𝑧 ∈ 𝐴 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝑇) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))
100	94, 95, 89, 60, 97, 99	opelopabf 4925	. . . 4 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑣⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑣 = 𝑇)} ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))
101	92, 100	bitri 263	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝑥 ∈ 𝐴 ↦ 𝑇) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))
102	88, 90, 101	3bitr4g 302	. 2 ⊢ (𝜑 → (⟨𝑧, 𝑤⟩ ∈ (𝐺 ∘ 𝐹) ↔ ⟨𝑧, 𝑤⟩ ∈ (𝑥 ∈ 𝐴 ↦ 𝑇)))
103	1, 4, 102	eqrelrdv 5139	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝑇))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695  Ⅎwnf 1699   ∈ wcel 1977  Ⅎwnfc 2738  ∀wral 2896  Vcvv 3173  ⦋csb 3499  ⟨cop 4131   class class class wbr 4583  {copab 4642   ↦ cmpt 4643  dom cdm 5038   ∘ ccom 5042  Rel wrel 5043  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812

This theorem is referenced by:  esumf1o  29439