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Theorem mapfien 9448

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.) (Revised by AV, 28-Jul-2024.)

**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 ∈ 𝑈)
mapfien.b	⊢ (𝜑 → 𝐵 ∈ 𝑉)
mapfien.c	⊢ (𝜑 → 𝐶 ∈ 𝑋)
mapfien.d	⊢ (𝜑 → 𝐷 ∈ 𝑌)
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) 𝑈(𝑥,𝑓) 𝑉(𝑥,𝑓) 𝑊(𝑓) 𝑋(𝑥,𝑓) 𝑌(𝑥,𝑓) 𝑍(𝑓)

Proof of Theorem mapfien Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2737	. 2 ⊢ (𝑓 ∈ 𝑆 ↦ (𝐺 ∘ (𝑓 ∘ 𝐹))) = (𝑓 ∈ 𝑆 ↦ (𝐺 ∘ (𝑓 ∘ 𝐹)))
2		mapfien.g	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷)
3		f1of 6848	. . . . . . 7 ⊢ (𝐺:𝐵–1-1-onto→𝐷 → 𝐺:𝐵⟶𝐷)
4	2, 3	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐺:𝐵⟶𝐷)
5	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐺:𝐵⟶𝐷)
6		breq1 5146	. . . . . . . . . 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 3695	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝑆 ↔ (𝑓 ∈ (𝐵 ↑_m 𝐴) ∧ 𝑓 finSupp 𝑍))
9	8	simplbi 497	. . . . . . . 8 ⊢ (𝑓 ∈ 𝑆 → 𝑓 ∈ (𝐵 ↑_m 𝐴))
10	9	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑓 ∈ (𝐵 ↑_m 𝐴))
11		elmapi 8889	. . . . . . 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 6848	. . . . . . . 8 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → 𝐹:𝐶⟶𝐴)
15	13, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐶⟶𝐴)
16	15	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐹:𝐶⟶𝐴)
17	12, 16	fcod 6761	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝑓 ∘ 𝐹):𝐶⟶𝐵)
18	5, 17	fcod 6761	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝐺 ∘ (𝑓 ∘ 𝐹)):𝐶⟶𝐷)
19		mapfien.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑌)
20		mapfien.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
21	19, 20	elmapd 8880	. . . . 5 ⊢ (𝜑 → ((𝐺 ∘ (𝑓 ∘ 𝐹)) ∈ (𝐷 ↑_m 𝐶) ↔ (𝐺 ∘ (𝑓 ∘ 𝐹)):𝐶⟶𝐷))
22	21	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → ((𝐺 ∘ (𝑓 ∘ 𝐹)) ∈ (𝐷 ↑_m 𝐶) ↔ (𝐺 ∘ (𝑓 ∘ 𝐹)):𝐶⟶𝐷))
23	18, 22	mpbird 257	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝐺 ∘ (𝑓 ∘ 𝐹)) ∈ (𝐷 ↑_m 𝐶))
24		mapfien.t	. . . 4 ⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑_m 𝐶) ∣ 𝑥 finSupp 𝑊}
25		mapfien.w	. . . 4 ⊢ 𝑊 = (𝐺‘𝑍)
26		mapfien.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
27		mapfien.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
28		mapfien.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
29	7, 24, 25, 13, 2, 26, 27, 20, 19, 28	mapfienlem1 9445	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝐺 ∘ (𝑓 ∘ 𝐹)) finSupp 𝑊)
30		breq1 5146	. . . 4 ⊢ (𝑥 = (𝐺 ∘ (𝑓 ∘ 𝐹)) → (𝑥 finSupp 𝑊 ↔ (𝐺 ∘ (𝑓 ∘ 𝐹)) finSupp 𝑊))
31	30, 24	elrab2 3695	. . 3 ⊢ ((𝐺 ∘ (𝑓 ∘ 𝐹)) ∈ 𝑇 ↔ ((𝐺 ∘ (𝑓 ∘ 𝐹)) ∈ (𝐷 ↑_m 𝐶) ∧ (𝐺 ∘ (𝑓 ∘ 𝐹)) finSupp 𝑊))
32	23, 29, 31	sylanbrc 583	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝐺 ∘ (𝑓 ∘ 𝐹)) ∈ 𝑇)
33	7, 24, 25, 13, 2, 26, 27, 20, 19, 28	mapfienlem3 9447	. 2 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) ∈ 𝑆)
34		coass 6285	. . . . . 6 ⊢ (((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) ∘ 𝐹) = ((^◡𝐺 ∘ 𝑔) ∘ (^◡𝐹 ∘ 𝐹))
35	13	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → 𝐹:𝐶–1-1-onto→𝐴)
36		f1ococnv1 6877	. . . . . . . . 9 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → (^◡𝐹 ∘ 𝐹) = ( I ↾ 𝐶))
37	35, 36	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → (^◡𝐹 ∘ 𝐹) = ( I ↾ 𝐶))
38	37	coeq2d 5873	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((^◡𝐺 ∘ 𝑔) ∘ (^◡𝐹 ∘ 𝐹)) = ((^◡𝐺 ∘ 𝑔) ∘ ( I ↾ 𝐶)))
39		f1ocnv 6860	. . . . . . . . . . . 12 ⊢ (𝐺:𝐵–1-1-onto→𝐷 → ^◡𝐺:𝐷–1-1-onto→𝐵)
40		f1of 6848	. . . . . . . . . . . 12 ⊢ (^◡𝐺:𝐷–1-1-onto→𝐵 → ^◡𝐺:𝐷⟶𝐵)
41	2, 39, 40	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ^◡𝐺:𝐷⟶𝐵)
42	41	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ^◡𝐺:𝐷⟶𝐵)
43		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → 𝑔 ∈ 𝑇)
44		breq1 5146	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑔 → (𝑥 finSupp 𝑊 ↔ 𝑔 finSupp 𝑊))
45	44, 24	elrab2 3695	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ 𝑇 ↔ (𝑔 ∈ (𝐷 ↑_m 𝐶) ∧ 𝑔 finSupp 𝑊))
46	43, 45	sylib 218	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → (𝑔 ∈ (𝐷 ↑_m 𝐶) ∧ 𝑔 finSupp 𝑊))
47	46	simpld 494	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → 𝑔 ∈ (𝐷 ↑_m 𝐶))
48		elmapi 8889	. . . . . . . . . . 11 ⊢ (𝑔 ∈ (𝐷 ↑_m 𝐶) → 𝑔:𝐶⟶𝐷)
49	47, 48	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → 𝑔:𝐶⟶𝐷)
50	42, 49	fcod 6761	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → (^◡𝐺 ∘ 𝑔):𝐶⟶𝐵)
51	50	adantrl 716	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → (^◡𝐺 ∘ 𝑔):𝐶⟶𝐵)
52		fcoi1 6782	. . . . . . . 8 ⊢ ((^◡𝐺 ∘ 𝑔):𝐶⟶𝐵 → ((^◡𝐺 ∘ 𝑔) ∘ ( I ↾ 𝐶)) = (^◡𝐺 ∘ 𝑔))
53	51, 52	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((^◡𝐺 ∘ 𝑔) ∘ ( I ↾ 𝐶)) = (^◡𝐺 ∘ 𝑔))
54	38, 53	eqtrd 2777	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((^◡𝐺 ∘ 𝑔) ∘ (^◡𝐹 ∘ 𝐹)) = (^◡𝐺 ∘ 𝑔))
55	34, 54	eqtrid 2789	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → (((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) ∘ 𝐹) = (^◡𝐺 ∘ 𝑔))
56	55	eqeq2d 2748	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((𝑓 ∘ 𝐹) = (((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) ∘ 𝐹) ↔ (𝑓 ∘ 𝐹) = (^◡𝐺 ∘ 𝑔)))
57		coass 6285	. . . . . . 7 ⊢ ((^◡𝐺 ∘ 𝐺) ∘ (𝑓 ∘ 𝐹)) = (^◡𝐺 ∘ (𝐺 ∘ (𝑓 ∘ 𝐹)))
58	2	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → 𝐺:𝐵–1-1-onto→𝐷)
59		f1ococnv1 6877	. . . . . . . . . 10 ⊢ (𝐺:𝐵–1-1-onto→𝐷 → (^◡𝐺 ∘ 𝐺) = ( I ↾ 𝐵))
60	58, 59	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → (^◡𝐺 ∘ 𝐺) = ( I ↾ 𝐵))
61	60	coeq1d 5872	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((^◡𝐺 ∘ 𝐺) ∘ (𝑓 ∘ 𝐹)) = (( I ↾ 𝐵) ∘ (𝑓 ∘ 𝐹)))
62	17	adantrr 717	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → (𝑓 ∘ 𝐹):𝐶⟶𝐵)
63		fcoi2 6783	. . . . . . . . 9 ⊢ ((𝑓 ∘ 𝐹):𝐶⟶𝐵 → (( I ↾ 𝐵) ∘ (𝑓 ∘ 𝐹)) = (𝑓 ∘ 𝐹))
64	62, 63	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → (( I ↾ 𝐵) ∘ (𝑓 ∘ 𝐹)) = (𝑓 ∘ 𝐹))
65	61, 64	eqtrd 2777	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((^◡𝐺 ∘ 𝐺) ∘ (𝑓 ∘ 𝐹)) = (𝑓 ∘ 𝐹))
66	57, 65	eqtr3id 2791	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → (^◡𝐺 ∘ (𝐺 ∘ (𝑓 ∘ 𝐹))) = (𝑓 ∘ 𝐹))
67	66	eqeq2d 2748	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((^◡𝐺 ∘ 𝑔) = (^◡𝐺 ∘ (𝐺 ∘ (𝑓 ∘ 𝐹))) ↔ (^◡𝐺 ∘ 𝑔) = (𝑓 ∘ 𝐹)))
68		eqcom 2744	. . . . 5 ⊢ ((^◡𝐺 ∘ 𝑔) = (𝑓 ∘ 𝐹) ↔ (𝑓 ∘ 𝐹) = (^◡𝐺 ∘ 𝑔))
69	67, 68	bitrdi 287	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((^◡𝐺 ∘ 𝑔) = (^◡𝐺 ∘ (𝐺 ∘ (𝑓 ∘ 𝐹))) ↔ (𝑓 ∘ 𝐹) = (^◡𝐺 ∘ 𝑔)))
70	56, 69	bitr4d 282	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((𝑓 ∘ 𝐹) = (((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) ∘ 𝐹) ↔ (^◡𝐺 ∘ 𝑔) = (^◡𝐺 ∘ (𝐺 ∘ (𝑓 ∘ 𝐹)))))
71		f1ofo 6855	. . . . 5 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → 𝐹:𝐶–onto→𝐴)
72	35, 71	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → 𝐹:𝐶–onto→𝐴)
73		ffn 6736	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 Fn 𝐴)
74	10, 11, 73	3syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑓 Fn 𝐴)
75	74	adantrr 717	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → 𝑓 Fn 𝐴)
76		f1ocnv 6860	. . . . . . . . 9 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → ^◡𝐹:𝐴–1-1-onto→𝐶)
77		f1of 6848	. . . . . . . . 9 ⊢ (^◡𝐹:𝐴–1-1-onto→𝐶 → ^◡𝐹:𝐴⟶𝐶)
78	13, 76, 77	3syl 18	. . . . . . . 8 ⊢ (𝜑 → ^◡𝐹:𝐴⟶𝐶)
79	78	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ^◡𝐹:𝐴⟶𝐶)
80	50, 79	fcod 6761	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹):𝐴⟶𝐵)
81	80	ffnd 6737	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) Fn 𝐴)
82	81	adantrl 716	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) Fn 𝐴)
83		cocan2 7312	. . . 4 ⊢ ((𝐹:𝐶–onto→𝐴 ∧ 𝑓 Fn 𝐴 ∧ ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) Fn 𝐴) → ((𝑓 ∘ 𝐹) = (((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) ∘ 𝐹) ↔ 𝑓 = ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹)))
84	72, 75, 82, 83	syl3anc 1373	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((𝑓 ∘ 𝐹) = (((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) ∘ 𝐹) ↔ 𝑓 = ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹)))
85	2, 39	syl 17	. . . . . 6 ⊢ (𝜑 → ^◡𝐺:𝐷–1-1-onto→𝐵)
86	85	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ^◡𝐺:𝐷–1-1-onto→𝐵)
87		f1of1 6847	. . . . 5 ⊢ (^◡𝐺:𝐷–1-1-onto→𝐵 → ^◡𝐺:𝐷–1-1→𝐵)
88	86, 87	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ^◡𝐺:𝐷–1-1→𝐵)
89	49	adantrl 716	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → 𝑔:𝐶⟶𝐷)
90	18	adantrr 717	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → (𝐺 ∘ (𝑓 ∘ 𝐹)):𝐶⟶𝐷)
91		cocan1 7311	. . . 4 ⊢ ((^◡𝐺:𝐷–1-1→𝐵 ∧ 𝑔:𝐶⟶𝐷 ∧ (𝐺 ∘ (𝑓 ∘ 𝐹)):𝐶⟶𝐷) → ((^◡𝐺 ∘ 𝑔) = (^◡𝐺 ∘ (𝐺 ∘ (𝑓 ∘ 𝐹))) ↔ 𝑔 = (𝐺 ∘ (𝑓 ∘ 𝐹))))
92	88, 89, 90, 91	syl3anc 1373	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((^◡𝐺 ∘ 𝑔) = (^◡𝐺 ∘ (𝐺 ∘ (𝑓 ∘ 𝐹))) ↔ 𝑔 = (𝐺 ∘ (𝑓 ∘ 𝐹))))
93	70, 84, 92	3bitr3d 309	. 2 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → (𝑓 = ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) ↔ 𝑔 = (𝐺 ∘ (𝑓 ∘ 𝐹))))
94	1, 32, 33, 93	f1o2d 7687	1 ⊢ (𝜑 → (𝑓 ∈ 𝑆 ↦ (𝐺 ∘ (𝑓 ∘ 𝐹))):𝑆–1-1-onto→𝑇)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 class class class wbr 5143 ↦ cmpt 5225 I cid 5577 ^◡ccnv 5684 ↾ cres 5687 ∘ ccom 5689 Fn wfn 6556 ⟶wf 6557 –1-1→wf1 6558 –onto→wfo 6559 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ↑_m cmap 8866 finSupp cfsupp 9401

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-1o 8506 df-map 8868 df-en 8986 df-dom 8987 df-fin 8989 df-fsupp 9402

This theorem is referenced by: mapfien2 9449 wemapwe 9737 oef1o 9738 fcobijfs 32734

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