Theorem mapfien 8313

Description: A bijection of the base sets induces a bijection on the set of finitely supported functions. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)

Hypotheses

Ref	Expression
mapfien.s	⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑𝑚 𝐴) ∣ 𝑥 finSupp 𝑍}
mapfien.t	⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑𝑚 𝐶) ∣ 𝑥 finSupp 𝑊}
mapfien.w	⊢ 𝑊 = (𝐺‘𝑍)
mapfien.f	⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)
mapfien.g	⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷)
mapfien.a	⊢ (𝜑 → 𝐴 ∈ V)
mapfien.b	⊢ (𝜑 → 𝐵 ∈ V)
mapfien.c	⊢ (𝜑 → 𝐶 ∈ V)
mapfien.d	⊢ (𝜑 → 𝐷 ∈ V)
mapfien.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)

Assertion

Ref	Expression
mapfien	⊢ (𝜑 → (𝑓 ∈ 𝑆 ↦ (𝐺 ∘ (𝑓 ∘ 𝐹))):𝑆–1-1-onto→𝑇)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝑓,𝐹   𝑓,𝐺,𝑥   𝜑,𝑓   𝑥,𝐷   𝑆,𝑓   𝑇,𝑓   𝑥,𝑊   𝑥,𝑍

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑓)   𝐵(𝑓)   𝐶(𝑓)   𝐷(𝑓)   𝑆(𝑥)   𝑇(𝑥)   𝑊(𝑓)   𝑍(𝑓)

Proof of Theorem mapfien

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 ⊢ (𝑓 ∈ 𝑆 ↦ (𝐺 ∘ (𝑓 ∘ 𝐹))) = (𝑓 ∈ 𝑆 ↦ (𝐺 ∘ (𝑓 ∘ 𝐹)))
2		mapfien.g	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷)
3		f1of 6137	. . . . . . 7 ⊢ (𝐺:𝐵–1-1-onto→𝐷 → 𝐺:𝐵⟶𝐷)
4	2, 3	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐺:𝐵⟶𝐷)
5	4	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐺:𝐵⟶𝐷)
6		breq1 4656	. . . . . . . . . 10 ⊢ (𝑥 = 𝑓 → (𝑥 finSupp 𝑍 ↔ 𝑓 finSupp 𝑍))
7		mapfien.s	. . . . . . . . . 10 ⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑𝑚 𝐴) ∣ 𝑥 finSupp 𝑍}
8	6, 7	elrab2 3366	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝑆 ↔ (𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑓 finSupp 𝑍))
9	8	simplbi 476	. . . . . . . 8 ⊢ (𝑓 ∈ 𝑆 → 𝑓 ∈ (𝐵 ↑𝑚 𝐴))
10	9	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑓 ∈ (𝐵 ↑𝑚 𝐴))
11		elmapi 7879	. . . . . . 7 ⊢ (𝑓 ∈ (𝐵 ↑𝑚 𝐴) → 𝑓:𝐴⟶𝐵)
12	10, 11	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑓:𝐴⟶𝐵)
13		mapfien.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)
14		f1of 6137	. . . . . . . 8 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → 𝐹:𝐶⟶𝐴)
15	13, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐶⟶𝐴)
16	15	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐹:𝐶⟶𝐴)
17		fco 6058	. . . . . 6 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝐹:𝐶⟶𝐴) → (𝑓 ∘ 𝐹):𝐶⟶𝐵)
18	12, 16, 17	syl2anc 693	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝑓 ∘ 𝐹):𝐶⟶𝐵)
19		fco 6058	. . . . 5 ⊢ ((𝐺:𝐵⟶𝐷 ∧ (𝑓 ∘ 𝐹):𝐶⟶𝐵) → (𝐺 ∘ (𝑓 ∘ 𝐹)):𝐶⟶𝐷)
20	5, 18, 19	syl2anc 693	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝐺 ∘ (𝑓 ∘ 𝐹)):𝐶⟶𝐷)
21		mapfien.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ V)
22		mapfien.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ V)
23	21, 22	elmapd 7871	. . . . 5 ⊢ (𝜑 → ((𝐺 ∘ (𝑓 ∘ 𝐹)) ∈ (𝐷 ↑𝑚 𝐶) ↔ (𝐺 ∘ (𝑓 ∘ 𝐹)):𝐶⟶𝐷))
24	23	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → ((𝐺 ∘ (𝑓 ∘ 𝐹)) ∈ (𝐷 ↑𝑚 𝐶) ↔ (𝐺 ∘ (𝑓 ∘ 𝐹)):𝐶⟶𝐷))
25	20, 24	mpbird 247	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝐺 ∘ (𝑓 ∘ 𝐹)) ∈ (𝐷 ↑𝑚 𝐶))
26		mapfien.t	. . . 4 ⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑𝑚 𝐶) ∣ 𝑥 finSupp 𝑊}
27		mapfien.w	. . . 4 ⊢ 𝑊 = (𝐺‘𝑍)
28		mapfien.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
29		mapfien.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ V)
30		mapfien.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
31	7, 26, 27, 13, 2, 28, 29, 22, 21, 30	mapfienlem1 8310	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝐺 ∘ (𝑓 ∘ 𝐹)) finSupp 𝑊)
32		breq1 4656	. . . 4 ⊢ (𝑥 = (𝐺 ∘ (𝑓 ∘ 𝐹)) → (𝑥 finSupp 𝑊 ↔ (𝐺 ∘ (𝑓 ∘ 𝐹)) finSupp 𝑊))
33	32, 26	elrab2 3366	. . 3 ⊢ ((𝐺 ∘ (𝑓 ∘ 𝐹)) ∈ 𝑇 ↔ ((𝐺 ∘ (𝑓 ∘ 𝐹)) ∈ (𝐷 ↑𝑚 𝐶) ∧ (𝐺 ∘ (𝑓 ∘ 𝐹)) finSupp 𝑊))
34	25, 31, 33	sylanbrc 698	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝐺 ∘ (𝑓 ∘ 𝐹)) ∈ 𝑇)
35	7, 26, 27, 13, 2, 28, 29, 22, 21, 30	mapfienlem3 8312	. 2 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) ∈ 𝑆)
36		coass 5654	. . . . . 6 ⊢ (((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) ∘ 𝐹) = ((◡𝐺 ∘ 𝑔) ∘ (◡𝐹 ∘ 𝐹))
37	13	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → 𝐹:𝐶–1-1-onto→𝐴)
38		f1ococnv1 6165	. . . . . . . . 9 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐶))
39	37, 38	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐶))
40	39	coeq2d 5284	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((◡𝐺 ∘ 𝑔) ∘ (◡𝐹 ∘ 𝐹)) = ((◡𝐺 ∘ 𝑔) ∘ ( I ↾ 𝐶)))
41		f1ocnv 6149	. . . . . . . . . . . 12 ⊢ (𝐺:𝐵–1-1-onto→𝐷 → ◡𝐺:𝐷–1-1-onto→𝐵)
42		f1of 6137	. . . . . . . . . . . 12 ⊢ (◡𝐺:𝐷–1-1-onto→𝐵 → ◡𝐺:𝐷⟶𝐵)
43	2, 41, 42	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ◡𝐺:𝐷⟶𝐵)
44	43	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ◡𝐺:𝐷⟶𝐵)
45		simpr 477	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → 𝑔 ∈ 𝑇)
46		breq1 4656	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑔 → (𝑥 finSupp 𝑊 ↔ 𝑔 finSupp 𝑊))
47	46, 26	elrab2 3366	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ 𝑇 ↔ (𝑔 ∈ (𝐷 ↑𝑚 𝐶) ∧ 𝑔 finSupp 𝑊))
48	45, 47	sylib 208	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → (𝑔 ∈ (𝐷 ↑𝑚 𝐶) ∧ 𝑔 finSupp 𝑊))
49	48	simpld 475	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → 𝑔 ∈ (𝐷 ↑𝑚 𝐶))
50		elmapi 7879	. . . . . . . . . . 11 ⊢ (𝑔 ∈ (𝐷 ↑𝑚 𝐶) → 𝑔:𝐶⟶𝐷)
51	49, 50	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → 𝑔:𝐶⟶𝐷)
52		fco 6058	. . . . . . . . . 10 ⊢ ((◡𝐺:𝐷⟶𝐵 ∧ 𝑔:𝐶⟶𝐷) → (◡𝐺 ∘ 𝑔):𝐶⟶𝐵)
53	44, 51, 52	syl2anc 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → (◡𝐺 ∘ 𝑔):𝐶⟶𝐵)
54	53	adantrl 752	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → (◡𝐺 ∘ 𝑔):𝐶⟶𝐵)
55		fcoi1 6078	. . . . . . . 8 ⊢ ((◡𝐺 ∘ 𝑔):𝐶⟶𝐵 → ((◡𝐺 ∘ 𝑔) ∘ ( I ↾ 𝐶)) = (◡𝐺 ∘ 𝑔))
56	54, 55	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((◡𝐺 ∘ 𝑔) ∘ ( I ↾ 𝐶)) = (◡𝐺 ∘ 𝑔))
57	40, 56	eqtrd 2656	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((◡𝐺 ∘ 𝑔) ∘ (◡𝐹 ∘ 𝐹)) = (◡𝐺 ∘ 𝑔))
58	36, 57	syl5eq 2668	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → (((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) ∘ 𝐹) = (◡𝐺 ∘ 𝑔))
59	58	eqeq2d 2632	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((𝑓 ∘ 𝐹) = (((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) ∘ 𝐹) ↔ (𝑓 ∘ 𝐹) = (◡𝐺 ∘ 𝑔)))
60		coass 5654	. . . . . . 7 ⊢ ((◡𝐺 ∘ 𝐺) ∘ (𝑓 ∘ 𝐹)) = (◡𝐺 ∘ (𝐺 ∘ (𝑓 ∘ 𝐹)))
61	2	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → 𝐺:𝐵–1-1-onto→𝐷)
62		f1ococnv1 6165	. . . . . . . . . 10 ⊢ (𝐺:𝐵–1-1-onto→𝐷 → (◡𝐺 ∘ 𝐺) = ( I ↾ 𝐵))
63	61, 62	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → (◡𝐺 ∘ 𝐺) = ( I ↾ 𝐵))
64	63	coeq1d 5283	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((◡𝐺 ∘ 𝐺) ∘ (𝑓 ∘ 𝐹)) = (( I ↾ 𝐵) ∘ (𝑓 ∘ 𝐹)))
65	18	adantrr 753	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → (𝑓 ∘ 𝐹):𝐶⟶𝐵)
66		fcoi2 6079	. . . . . . . . 9 ⊢ ((𝑓 ∘ 𝐹):𝐶⟶𝐵 → (( I ↾ 𝐵) ∘ (𝑓 ∘ 𝐹)) = (𝑓 ∘ 𝐹))
67	65, 66	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → (( I ↾ 𝐵) ∘ (𝑓 ∘ 𝐹)) = (𝑓 ∘ 𝐹))
68	64, 67	eqtrd 2656	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((◡𝐺 ∘ 𝐺) ∘ (𝑓 ∘ 𝐹)) = (𝑓 ∘ 𝐹))
69	60, 68	syl5eqr 2670	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → (◡𝐺 ∘ (𝐺 ∘ (𝑓 ∘ 𝐹))) = (𝑓 ∘ 𝐹))
70	69	eqeq2d 2632	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((◡𝐺 ∘ 𝑔) = (◡𝐺 ∘ (𝐺 ∘ (𝑓 ∘ 𝐹))) ↔ (◡𝐺 ∘ 𝑔) = (𝑓 ∘ 𝐹)))
71		eqcom 2629	. . . . 5 ⊢ ((◡𝐺 ∘ 𝑔) = (𝑓 ∘ 𝐹) ↔ (𝑓 ∘ 𝐹) = (◡𝐺 ∘ 𝑔))
72	70, 71	syl6bb 276	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((◡𝐺 ∘ 𝑔) = (◡𝐺 ∘ (𝐺 ∘ (𝑓 ∘ 𝐹))) ↔ (𝑓 ∘ 𝐹) = (◡𝐺 ∘ 𝑔)))
73	59, 72	bitr4d 271	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((𝑓 ∘ 𝐹) = (((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) ∘ 𝐹) ↔ (◡𝐺 ∘ 𝑔) = (◡𝐺 ∘ (𝐺 ∘ (𝑓 ∘ 𝐹)))))
74		f1ofo 6144	. . . . 5 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → 𝐹:𝐶–onto→𝐴)
75	37, 74	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → 𝐹:𝐶–onto→𝐴)
76		ffn 6045	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 Fn 𝐴)
77	10, 11, 76	3syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑓 Fn 𝐴)
78	77	adantrr 753	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → 𝑓 Fn 𝐴)
79		f1ocnv 6149	. . . . . . . . 9 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → ◡𝐹:𝐴–1-1-onto→𝐶)
80		f1of 6137	. . . . . . . . 9 ⊢ (◡𝐹:𝐴–1-1-onto→𝐶 → ◡𝐹:𝐴⟶𝐶)
81	13, 79, 80	3syl 18	. . . . . . . 8 ⊢ (𝜑 → ◡𝐹:𝐴⟶𝐶)
82	81	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ◡𝐹:𝐴⟶𝐶)
83		fco 6058	. . . . . . 7 ⊢ (((◡𝐺 ∘ 𝑔):𝐶⟶𝐵 ∧ ◡𝐹:𝐴⟶𝐶) → ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹):𝐴⟶𝐵)
84	53, 82, 83	syl2anc 693	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹):𝐴⟶𝐵)
85		ffn 6045	. . . . . 6 ⊢ (((◡𝐺 ∘ 𝑔) ∘ ◡𝐹):𝐴⟶𝐵 → ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) Fn 𝐴)
86	84, 85	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) Fn 𝐴)
87	86	adantrl 752	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) Fn 𝐴)
88		cocan2 6547	. . . 4 ⊢ ((𝐹:𝐶–onto→𝐴 ∧ 𝑓 Fn 𝐴 ∧ ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) Fn 𝐴) → ((𝑓 ∘ 𝐹) = (((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) ∘ 𝐹) ↔ 𝑓 = ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹)))
89	75, 78, 87, 88	syl3anc 1326	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((𝑓 ∘ 𝐹) = (((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) ∘ 𝐹) ↔ 𝑓 = ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹)))
90	2, 41	syl 17	. . . . . 6 ⊢ (𝜑 → ◡𝐺:𝐷–1-1-onto→𝐵)
91	90	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ◡𝐺:𝐷–1-1-onto→𝐵)
92		f1of1 6136	. . . . 5 ⊢ (◡𝐺:𝐷–1-1-onto→𝐵 → ◡𝐺:𝐷–1-1→𝐵)
93	91, 92	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ◡𝐺:𝐷–1-1→𝐵)
94	51	adantrl 752	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → 𝑔:𝐶⟶𝐷)
95	20	adantrr 753	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → (𝐺 ∘ (𝑓 ∘ 𝐹)):𝐶⟶𝐷)
96		cocan1 6546	. . . 4 ⊢ ((◡𝐺:𝐷–1-1→𝐵 ∧ 𝑔:𝐶⟶𝐷 ∧ (𝐺 ∘ (𝑓 ∘ 𝐹)):𝐶⟶𝐷) → ((◡𝐺 ∘ 𝑔) = (◡𝐺 ∘ (𝐺 ∘ (𝑓 ∘ 𝐹))) ↔ 𝑔 = (𝐺 ∘ (𝑓 ∘ 𝐹))))
97	93, 94, 95, 96	syl3anc 1326	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((◡𝐺 ∘ 𝑔) = (◡𝐺 ∘ (𝐺 ∘ (𝑓 ∘ 𝐹))) ↔ 𝑔 = (𝐺 ∘ (𝑓 ∘ 𝐹))))
98	73, 89, 97	3bitr3d 298	. 2 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → (𝑓 = ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) ↔ 𝑔 = (𝐺 ∘ (𝑓 ∘ 𝐹))))
99	1, 34, 35, 98	f1o2d 6887	1 ⊢ (𝜑 → (𝑓 ∈ 𝑆 ↦ (𝐺 ∘ (𝑓 ∘ 𝐹))):𝑆–1-1-onto→𝑇)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {crab 2916  Vcvv 3200   class class class wbr 4653   ↦ cmpt 4729   I cid 5023  ^◡ccnv 5113   ↾ cres 5116   ∘ ccom 5118   Fn wfn 5883  ⟶wf 5884  –1-1→wf1 5885  –onto→wfo 5886  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857   finSupp cfsupp 8275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-1o 7560  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-fin 7959  df-fsupp 8276

This theorem is referenced by:  mapfien2  8314  wemapwe  8594  oef1o  8595  fcobijfs  29501