Theorem fmptco 5848

Description: Composition of two functions expressed as ordered-pair class abstractions. If 𝐹 has the equation ( x + 2 ) and 𝐺 the equation ( 3 * z ) then (𝐺 ∘ 𝐹) has the equation ( 3 * ( x + 2 ) ) . (Contributed by FL, 21-Jun-2012.) (Revised by Mario Carneiro, 24-Jul-2014.)

Hypotheses

Ref	Expression
fmptco.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ 𝐵)
fmptco.2	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑅))
fmptco.3	⊢ (𝜑 → 𝐺 = (𝑦 ∈ 𝐵 ↦ 𝑆))
fmptco.4	⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵   𝑦,𝑅   𝜑,𝑥   𝑥,𝑆   𝑦,𝑇

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

Proof of Theorem fmptco

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

Step	Hyp	Ref	Expression
1		relco 5266	. 2 ⊢ Rel (𝐺 ∘ 𝐹)
2		funmpt 5395	. . 3 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝑇)
3		funrel 5374	. . 3 ⊢ (Fun (𝑥 ∈ 𝐴 ↦ 𝑇) → Rel (𝑥 ∈ 𝐴 ↦ 𝑇))
4	2, 3	ax-mp 5	. 2 ⊢ Rel (𝑥 ∈ 𝐴 ↦ 𝑇)
5		fmptco.1	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ 𝐵)
6		eqid 2234	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑅) = (𝑥 ∈ 𝐴 ↦ 𝑅)
7	5, 6	fmptd 5836	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑅):𝐴⟶𝐵)
8		fmptco.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑅))
9	8	feq1d 5500	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ (𝑥 ∈ 𝐴 ↦ 𝑅):𝐴⟶𝐵))
10	7, 9	mpbird 167	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
11		ffun 5516	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
12	10, 11	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → Fun 𝐹)
13		funbrfv 5718	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → (𝑧𝐹𝑢 → (𝐹‘𝑧) = 𝑢))
14	13	imp 124	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑧𝐹𝑢) → (𝐹‘𝑧) = 𝑢)
15	12, 14	sylan 283	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧𝐹𝑢) → (𝐹‘𝑧) = 𝑢)
16	15	eqcomd 2240	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧𝐹𝑢) → 𝑢 = (𝐹‘𝑧))
17	16	a1d 22	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧𝐹𝑢) → (𝑢𝐺𝑤 → 𝑢 = (𝐹‘𝑧)))
18	17	expimpd 363	. . . . . 6 ⊢ (𝜑 → ((𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤) → 𝑢 = (𝐹‘𝑧)))
19	18	pm4.71rd 394	. . . . 5 ⊢ (𝜑 → ((𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤) ↔ (𝑢 = (𝐹‘𝑧) ∧ (𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤))))
20	19	exbidv 1874	. . . 4 ⊢ (𝜑 → (∃𝑢(𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤) ↔ ∃𝑢(𝑢 = (𝐹‘𝑧) ∧ (𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤))))
21		exsimpl 1666	. . . . . . 7 ⊢ (∃𝑢(𝑢 = (𝐹‘𝑧) ∧ (𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤)) → ∃𝑢 𝑢 = (𝐹‘𝑧))
22		isset 2822	. . . . . . 7 ⊢ ((𝐹‘𝑧) ∈ V ↔ ∃𝑢 𝑢 = (𝐹‘𝑧))
23	21, 22	sylibr 134	. . . . . 6 ⊢ (∃𝑢(𝑢 = (𝐹‘𝑧) ∧ (𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤)) → (𝐹‘𝑧) ∈ V)
24	23	a1i 9	. . . . 5 ⊢ (𝜑 → (∃𝑢(𝑢 = (𝐹‘𝑧) ∧ (𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤)) → (𝐹‘𝑧) ∈ V))
25	12	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → Fun 𝐹)
26		fdm 5519	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
27	10, 26	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐹 = 𝐴)
28	27	eleq2d 2304	. . . . . . . . 9 ⊢ (𝜑 → (𝑧 ∈ dom 𝐹 ↔ 𝑧 ∈ 𝐴))
29	28	biimpar 297	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ dom 𝐹)
30		funfvex 5692	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) ∈ V)
31	25, 29, 30	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ V)
32	31	adantrr 479	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)) → (𝐹‘𝑧) ∈ V)
33	32	ex 115	. . . . 5 ⊢ (𝜑 → ((𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇) → (𝐹‘𝑧) ∈ V))
34		breq2 4118	. . . . . . . . 9 ⊢ (𝑢 = (𝐹‘𝑧) → (𝑧𝐹𝑢 ↔ 𝑧𝐹(𝐹‘𝑧)))
35		breq1 4117	. . . . . . . . 9 ⊢ (𝑢 = (𝐹‘𝑧) → (𝑢𝐺𝑤 ↔ (𝐹‘𝑧)𝐺𝑤))
36	34, 35	anbi12d 473	. . . . . . . 8 ⊢ (𝑢 = (𝐹‘𝑧) → ((𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤) ↔ (𝑧𝐹(𝐹‘𝑧) ∧ (𝐹‘𝑧)𝐺𝑤)))
37	36	ceqsexgv 2949	. . . . . . 7 ⊢ ((𝐹‘𝑧) ∈ V → (∃𝑢(𝑢 = (𝐹‘𝑧) ∧ (𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤)) ↔ (𝑧𝐹(𝐹‘𝑧) ∧ (𝐹‘𝑧)𝐺𝑤)))
38		funfvbrb 5796	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → (𝑧 ∈ dom 𝐹 ↔ 𝑧𝐹(𝐹‘𝑧)))
39	12, 38	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ dom 𝐹 ↔ 𝑧𝐹(𝐹‘𝑧)))
40	39, 28	bitr3d 190	. . . . . . . . 9 ⊢ (𝜑 → (𝑧𝐹(𝐹‘𝑧) ↔ 𝑧 ∈ 𝐴))
41	8	fveq1d 5677	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘𝑧) = ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧))
42		fmptco.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 = (𝑦 ∈ 𝐵 ↦ 𝑆))
43		eqidd 2235	. . . . . . . . . 10 ⊢ (𝜑 → 𝑤 = 𝑤)
44	41, 42, 43	breq123d 4128	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹‘𝑧)𝐺𝑤 ↔ ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤))
45	40, 44	anbi12d 473	. . . . . . . 8 ⊢ (𝜑 → ((𝑧𝐹(𝐹‘𝑧) ∧ (𝐹‘𝑧)𝐺𝑤) ↔ (𝑧 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤)))
46		nfcv 2386	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑧
47		nfv 1577	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝜑
48		nffvmpt1 5686	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)
49		nfcv 2386	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐵 ↦ 𝑆)
50		nfcv 2386	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥𝑤
51	48, 49, 50	nfbr 4161	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤
52		nfcsb1v 3174	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝑇
53	52	nfeq2 2398	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝑤 = ⦋𝑧 / 𝑥⦌𝑇
54	51, 53	nfbi 1638	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)
55	47, 54	nfim 1621	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝜑 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))
56		fveq2 5675	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧))
57	56	breq1d 4124	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤))
58		csbeq1a 3150	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → 𝑇 = ⦋𝑧 / 𝑥⦌𝑇)
59	58	eqeq2d 2246	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑤 = 𝑇 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))
60	57, 59	bibi12d 235	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ((((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = 𝑇) ↔ (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))
61	60	imbi2d 230	. . . . . . . . . . 11 ⊢ (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))))
62		vex 2818	. . . . . . . . . . . . . 14 ⊢ 𝑤 ∈ V
63		simpl 109	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 = 𝑅 ∧ 𝑢 = 𝑤) → 𝑦 = 𝑅)
64	63	eleq1d 2303	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 = 𝑅 ∧ 𝑢 = 𝑤) → (𝑦 ∈ 𝐵 ↔ 𝑅 ∈ 𝐵))
65		simpr 110	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 = 𝑅 ∧ 𝑢 = 𝑤) → 𝑢 = 𝑤)
66		fmptco.4	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇)
67	66	adantr 276	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 = 𝑅 ∧ 𝑢 = 𝑤) → 𝑆 = 𝑇)
68	65, 67	eqeq12d 2249	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 = 𝑅 ∧ 𝑢 = 𝑤) → (𝑢 = 𝑆 ↔ 𝑤 = 𝑇))
69	64, 68	anbi12d 473	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 = 𝑅 ∧ 𝑢 = 𝑤) → ((𝑦 ∈ 𝐵 ∧ 𝑢 = 𝑆) ↔ (𝑅 ∈ 𝐵 ∧ 𝑤 = 𝑇)))
70		df-mpt 4178	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝐵 ↦ 𝑆) = {⟨𝑦, 𝑢⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑢 = 𝑆)}
71	69, 70	brabga 4387	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ 𝐵 ∧ 𝑤 ∈ V) → (𝑅(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ (𝑅 ∈ 𝐵 ∧ 𝑤 = 𝑇)))
72	5, 62, 71	sylancl 413	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ (𝑅 ∈ 𝐵 ∧ 𝑤 = 𝑇)))
73		simpr 110	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
74	6	fvmpt2 5766	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑅 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥) = 𝑅)
75	73, 5, 74	syl2anc 411	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥) = 𝑅)
76	75	breq1d 4124	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑅(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤))
77	5	biantrurd 305	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑤 = 𝑇 ↔ (𝑅 ∈ 𝐵 ∧ 𝑤 = 𝑇)))
78	72, 76, 77	3bitr4d 220	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = 𝑇))
79	78	expcom 116	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑥)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = 𝑇)))
80	46, 55, 61, 79	vtoclgaf 2882	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝐴 → (𝜑 → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))
81	80	impcom 125	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))
82	81	pm5.32da 452	. . . . . . . 8 ⊢ (𝜑 → ((𝑧 ∈ 𝐴 ∧ ((𝑥 ∈ 𝐴 ↦ 𝑅)‘𝑧)(𝑦 ∈ 𝐵 ↦ 𝑆)𝑤) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))
83	45, 82	bitrd 188	. . . . . . 7 ⊢ (𝜑 → ((𝑧𝐹(𝐹‘𝑧) ∧ (𝐹‘𝑧)𝐺𝑤) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))
84	37, 83	sylan9bbr 463	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹‘𝑧) ∈ V) → (∃𝑢(𝑢 = (𝐹‘𝑧) ∧ (𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤)) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))
85	84	ex 115	. . . . 5 ⊢ (𝜑 → ((𝐹‘𝑧) ∈ V → (∃𝑢(𝑢 = (𝐹‘𝑧) ∧ (𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤)) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))))
86	24, 33, 85	pm5.21ndd 713	. . . 4 ⊢ (𝜑 → (∃𝑢(𝑢 = (𝐹‘𝑧) ∧ (𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤)) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))
87	20, 86	bitrd 188	. . 3 ⊢ (𝜑 → (∃𝑢(𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))
88		vex 2818	. . . 4 ⊢ 𝑧 ∈ V
89	88, 62	opelco 4932	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝐺 ∘ 𝐹) ↔ ∃𝑢(𝑧𝐹𝑢 ∧ 𝑢𝐺𝑤))
90		df-mpt 4178	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑇) = {⟨𝑥, 𝑣⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑣 = 𝑇)}
91	90	eleq2i 2301	. . . 4 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝑥 ∈ 𝐴 ↦ 𝑇) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑣⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑣 = 𝑇)})
92		nfv 1577	. . . . . 6 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
93	52	nfeq2 2398	. . . . . 6 ⊢ Ⅎ𝑥 𝑣 = ⦋𝑧 / 𝑥⦌𝑇
94	92, 93	nfan 1614	. . . . 5 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝑇)
95		nfv 1577	. . . . 5 ⊢ Ⅎ𝑣(𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)
96		eleq1 2297	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
97	58	eqeq2d 2246	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑣 = 𝑇 ↔ 𝑣 = ⦋𝑧 / 𝑥⦌𝑇))
98	96, 97	anbi12d 473	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝑣 = 𝑇) ↔ (𝑧 ∈ 𝐴 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝑇)))
99		eqeq1 2241	. . . . . 6 ⊢ (𝑣 = 𝑤 → (𝑣 = ⦋𝑧 / 𝑥⦌𝑇 ↔ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))
100	99	anbi2d 464	. . . . 5 ⊢ (𝑣 = 𝑤 → ((𝑧 ∈ 𝐴 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝑇) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇)))
101	94, 95, 88, 62, 98, 100	opelopabf 4398	. . . 4 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑣⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑣 = 𝑇)} ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))
102	91, 101	bitri 184	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝑥 ∈ 𝐴 ↦ 𝑇) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 = ⦋𝑧 / 𝑥⦌𝑇))
103	87, 89, 102	3bitr4g 223	. 2 ⊢ (𝜑 → (⟨𝑧, 𝑤⟩ ∈ (𝐺 ∘ 𝐹) ↔ ⟨𝑧, 𝑤⟩ ∈ (𝑥 ∈ 𝐴 ↦ 𝑇)))
104	1, 4, 103	eqrelrdv 4851	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝑇))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398  ∃wex 1541   ∈ wcel 2205  Vcvv 2815  ⦋csb 3141  ⟨cop 3697   class class class wbr 4114  {copab 4175   ↦ cmpt 4176  dom cdm 4754   ∘ ccom 4758  Rel wrel 4759  Fun wfun 5351  ⟶wf 5353  ‘cfv 5357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365

This theorem is referenced by:  fmptcof  5849  cofmpt  5851  fcompt  5852  fcoconst  5853  ofco  6294  gsumfzmhm2  14097  prdsidlem  14135  pws0g  14155  pwsinvg  14157  pwssub  14158  psrlinv  14965  lmcn2  15271  cdivcncfap  15595  negfcncf  15597  dvcj  15700  dvfre  15701  dvmptcjx  15715  plyco  15750  plycjlemc  15751