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Theorem mapfien 9437

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⟶1-1onto𝐴)
mapfien.g	⊢ (𝜑 → 𝐺:𝐵⟶1-1onto𝐷)
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⟶1-1onto𝑇)

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 2763	. 2 ⊢ (𝑓 ∈ 𝑆 ↦ (𝐺 ∘ (𝑓 ∘ 𝐹))) = (𝑓 ∈ 𝑆 ↦ (𝐺 ∘ (𝑓 ∘ 𝐹)))
2		mapfien.g	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝐵⟶1-1onto𝐷)
3		f1of 6854	. . . . . . 7 ⊢ (𝐺:𝐵⟶1-1onto𝐷 → 𝐺:𝐵⟶𝐷)
4	2, 3	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐺:𝐵⟶𝐷)
5	4	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐺:𝐵⟶𝐷)
6		breq1 5111	. . . . . . . . . 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 3652	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝑆 ↔ (𝑓 ∈ (𝐵 ↑_m 𝐴) ∧ 𝑓 finSupp 𝑍))
9	8	simplbi 502	. . . . . . . 8 ⊢ (𝑓 ∈ 𝑆 → 𝑓 ∈ (𝐵 ↑_m 𝐴))
10	9	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑓 ∈ (𝐵 ↑_m 𝐴))
11		elmapi 8907	. . . . . . 7 ⊢ (𝑓 ∈ (𝐵 ↑_m 𝐴) → 𝑓:𝐴⟶𝐵)
12	10, 11	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑓:𝐴⟶𝐵)
13		mapfien.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐶⟶1-1onto𝐴)
14		f1of 6854	. . . . . . . 8 ⊢ (𝐹:𝐶⟶1-1onto𝐴 → 𝐹:𝐶⟶𝐴)
15	13, 14	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐶⟶𝐴)
16	15	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝐹:𝐶⟶𝐴)
17	12, 16	fcod 6765	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝑓 ∘ 𝐹):𝐶⟶𝐵)
18	5, 17	fcod 6765	. . . 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 8898	. . . . 5 ⊢ (𝜑 → ((𝐺 ∘ (𝑓 ∘ 𝐹)) ∈ (𝐷 ↑_m 𝐶) ↔ (𝐺 ∘ (𝑓 ∘ 𝐹)):𝐶⟶𝐷))
22	21	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → ((𝐺 ∘ (𝑓 ∘ 𝐹)) ∈ (𝐷 ↑_m 𝐶) ↔ (𝐺 ∘ (𝑓 ∘ 𝐹)):𝐶⟶𝐷))
23	18, 22	mpbird 260	. . 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 9434	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝐺 ∘ (𝑓 ∘ 𝐹)) finSupp 𝑊)
30		breq1 5111	. . . 4 ⊢ (𝑥 = (𝐺 ∘ (𝑓 ∘ 𝐹)) → (𝑥 finSupp 𝑊 ↔ (𝐺 ∘ (𝑓 ∘ 𝐹)) finSupp 𝑊))
31	30, 24	elrab2 3652	. . 3 ⊢ ((𝐺 ∘ (𝑓 ∘ 𝐹)) ∈ 𝑇 ↔ ((𝐺 ∘ (𝑓 ∘ 𝐹)) ∈ (𝐷 ↑_m 𝐶) ∧ (𝐺 ∘ (𝑓 ∘ 𝐹)) finSupp 𝑊))
32	23, 29, 31	sylanbrc 595	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝐺 ∘ (𝑓 ∘ 𝐹)) ∈ 𝑇)
33	7, 24, 25, 13, 2, 26, 27, 20, 19, 28	mapfienlem3 9436	. 2 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) ∈ 𝑆)
34		coass 6290	. . . . . 6 ⊢ (((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) ∘ 𝐹) = ((^◡𝐺 ∘ 𝑔) ∘ (^◡𝐹 ∘ 𝐹))
35	13	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → 𝐹:𝐶⟶1-1onto𝐴)
36		f1ococnv1 6885	. . . . . . . . 9 ⊢ (𝐹:𝐶⟶1-1onto𝐴 → (^◡𝐹 ∘ 𝐹) = ( I ↾ 𝐶))
37	35, 36	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → (^◡𝐹 ∘ 𝐹) = ( I ↾ 𝐶))
38	37	coeq2d 5851	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((^◡𝐺 ∘ 𝑔) ∘ (^◡𝐹 ∘ 𝐹)) = ((^◡𝐺 ∘ 𝑔) ∘ ( I ↾ 𝐶)))
39		f1ocnv 6867	. . . . . . . . . . . 12 ⊢ (𝐺:𝐵⟶1-1onto𝐷 → ^◡𝐺:𝐷⟶1-1onto𝐵)
40		f1of 6854	. . . . . . . . . . . 12 ⊢ (^◡𝐺:𝐷⟶1-1onto𝐵 → ^◡𝐺:𝐷⟶𝐵)
41	2, 39, 40	3syl 19	. . . . . . . . . . 11 ⊢ (𝜑 → ^◡𝐺:𝐷⟶𝐵)
42	41	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ^◡𝐺:𝐷⟶𝐵)
43		breq1 5111	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑔 → (𝑥 finSupp 𝑊 ↔ 𝑔 finSupp 𝑊))
44	43, 24	elrab2 3652	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ 𝑇 ↔ (𝑔 ∈ (𝐷 ↑_m 𝐶) ∧ 𝑔 finSupp 𝑊))
45	44	bilani 510	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → (𝑔 ∈ (𝐷 ↑_m 𝐶) ∧ 𝑔 finSupp 𝑊))
46	45	simpld 500	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → 𝑔 ∈ (𝐷 ↑_m 𝐶))
47		elmapi 8907	. . . . . . . . . . 11 ⊢ (𝑔 ∈ (𝐷 ↑_m 𝐶) → 𝑔:𝐶⟶𝐷)
48	46, 47	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → 𝑔:𝐶⟶𝐷)
49	42, 48	fcod 6765	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → (^◡𝐺 ∘ 𝑔):𝐶⟶𝐵)
50	49	adantrl 729	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → (^◡𝐺 ∘ 𝑔):𝐶⟶𝐵)
51		fcoi1 6786	. . . . . . . 8 ⊢ ((^◡𝐺 ∘ 𝑔):𝐶⟶𝐵 → ((^◡𝐺 ∘ 𝑔) ∘ ( I ↾ 𝐶)) = (^◡𝐺 ∘ 𝑔))
52	50, 51	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((^◡𝐺 ∘ 𝑔) ∘ ( I ↾ 𝐶)) = (^◡𝐺 ∘ 𝑔))
53	38, 52	eqtrd 2798	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((^◡𝐺 ∘ 𝑔) ∘ (^◡𝐹 ∘ 𝐹)) = (^◡𝐺 ∘ 𝑔))
54	34, 53	eqtrid 2810	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → (((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) ∘ 𝐹) = (^◡𝐺 ∘ 𝑔))
55	54	eqeq2d 2774	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((𝑓 ∘ 𝐹) = (((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) ∘ 𝐹) ↔ (𝑓 ∘ 𝐹) = (^◡𝐺 ∘ 𝑔)))
56		coass 6290	. . . . . . 7 ⊢ ((^◡𝐺 ∘ 𝐺) ∘ (𝑓 ∘ 𝐹)) = (^◡𝐺 ∘ (𝐺 ∘ (𝑓 ∘ 𝐹)))
57	2	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → 𝐺:𝐵⟶1-1onto𝐷)
58		f1ococnv1 6885	. . . . . . . . . 10 ⊢ (𝐺:𝐵⟶1-1onto𝐷 → (^◡𝐺 ∘ 𝐺) = ( I ↾ 𝐵))
59	57, 58	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → (^◡𝐺 ∘ 𝐺) = ( I ↾ 𝐵))
60	59	coeq1d 5850	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((^◡𝐺 ∘ 𝐺) ∘ (𝑓 ∘ 𝐹)) = (( I ↾ 𝐵) ∘ (𝑓 ∘ 𝐹)))
61	17	adantrr 730	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → (𝑓 ∘ 𝐹):𝐶⟶𝐵)
62		fcoi2 6787	. . . . . . . . 9 ⊢ ((𝑓 ∘ 𝐹):𝐶⟶𝐵 → (( I ↾ 𝐵) ∘ (𝑓 ∘ 𝐹)) = (𝑓 ∘ 𝐹))
63	61, 62	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → (( I ↾ 𝐵) ∘ (𝑓 ∘ 𝐹)) = (𝑓 ∘ 𝐹))
64	60, 63	eqtrd 2798	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((^◡𝐺 ∘ 𝐺) ∘ (𝑓 ∘ 𝐹)) = (𝑓 ∘ 𝐹))
65	56, 64	eqtr3id 2812	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → (^◡𝐺 ∘ (𝐺 ∘ (𝑓 ∘ 𝐹))) = (𝑓 ∘ 𝐹))
66	65	eqeq2d 2774	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((^◡𝐺 ∘ 𝑔) = (^◡𝐺 ∘ (𝐺 ∘ (𝑓 ∘ 𝐹))) ↔ (^◡𝐺 ∘ 𝑔) = (𝑓 ∘ 𝐹)))
67		eqcom 2770	. . . . 5 ⊢ ((^◡𝐺 ∘ 𝑔) = (𝑓 ∘ 𝐹) ↔ (𝑓 ∘ 𝐹) = (^◡𝐺 ∘ 𝑔))
68	66, 67	bitrdi 290	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((^◡𝐺 ∘ 𝑔) = (^◡𝐺 ∘ (𝐺 ∘ (𝑓 ∘ 𝐹))) ↔ (𝑓 ∘ 𝐹) = (^◡𝐺 ∘ 𝑔)))
69	55, 68	bitr4d 285	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((𝑓 ∘ 𝐹) = (((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) ∘ 𝐹) ↔ (^◡𝐺 ∘ 𝑔) = (^◡𝐺 ∘ (𝐺 ∘ (𝑓 ∘ 𝐹)))))
70		f1ofo 6862	. . . . 5 ⊢ (𝐹:𝐶⟶1-1onto𝐴 → 𝐹:𝐶⟶onto𝐴)
71	35, 70	syl 18	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → 𝐹:𝐶⟶onto𝐴)
72		ffn 6739	. . . . . 6 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 Fn 𝐴)
73	10, 11, 72	3syl 19	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑆) → 𝑓 Fn 𝐴)
74	73	adantrr 730	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → 𝑓 Fn 𝐴)
75		f1ocnv 6867	. . . . . . . . 9 ⊢ (𝐹:𝐶⟶1-1onto𝐴 → ^◡𝐹:𝐴⟶1-1onto𝐶)
76		f1of 6854	. . . . . . . . 9 ⊢ (^◡𝐹:𝐴⟶1-1onto𝐶 → ^◡𝐹:𝐴⟶𝐶)
77	13, 75, 76	3syl 19	. . . . . . . 8 ⊢ (𝜑 → ^◡𝐹:𝐴⟶𝐶)
78	77	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ^◡𝐹:𝐴⟶𝐶)
79	49, 78	fcod 6765	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹):𝐴⟶𝐵)
80	79	ffnd 6740	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) Fn 𝐴)
81	80	adantrl 729	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) Fn 𝐴)
82		cocan2 7331	. . . 4 ⊢ ((𝐹:𝐶⟶onto𝐴 ∧ 𝑓 Fn 𝐴 ∧ ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) Fn 𝐴) → ((𝑓 ∘ 𝐹) = (((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) ∘ 𝐹) ↔ 𝑓 = ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹)))
83	71, 74, 81, 82	syl3anc 1399	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((𝑓 ∘ 𝐹) = (((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) ∘ 𝐹) ↔ 𝑓 = ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹)))
84	2, 39	syl 18	. . . . . 6 ⊢ (𝜑 → ^◡𝐺:𝐷⟶1-1onto𝐵)
85	84	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ^◡𝐺:𝐷⟶1-1onto𝐵)
86		f1of1 6853	. . . . 5 ⊢ (^◡𝐺:𝐷⟶1-1onto𝐵 → ^◡𝐺:𝐷⟶1-1𝐵)
87	85, 86	syl 18	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ^◡𝐺:𝐷⟶1-1𝐵)
88	48	adantrl 729	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → 𝑔:𝐶⟶𝐷)
89	18	adantrr 730	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → (𝐺 ∘ (𝑓 ∘ 𝐹)):𝐶⟶𝐷)
90		cocan1 7330	. . . 4 ⊢ ((^◡𝐺:𝐷⟶1-1𝐵 ∧ 𝑔:𝐶⟶𝐷 ∧ (𝐺 ∘ (𝑓 ∘ 𝐹)):𝐶⟶𝐷) → ((^◡𝐺 ∘ 𝑔) = (^◡𝐺 ∘ (𝐺 ∘ (𝑓 ∘ 𝐹))) ↔ 𝑔 = (𝐺 ∘ (𝑓 ∘ 𝐹))))
91	87, 88, 89, 90	syl3anc 1399	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → ((^◡𝐺 ∘ 𝑔) = (^◡𝐺 ∘ (𝐺 ∘ (𝑓 ∘ 𝐹))) ↔ 𝑔 = (𝐺 ∘ (𝑓 ∘ 𝐹))))
92	69, 83, 91	3bitr3d 312	. 2 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑇)) → (𝑓 = ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) ↔ 𝑔 = (𝐺 ∘ (𝑓 ∘ 𝐹))))
93	1, 32, 33, 92	f1o2d 7707	1 ⊢ (𝜑 → (𝑓 ∈ 𝑆 ↦ (𝐺 ∘ (𝑓 ∘ 𝐹))):𝑆⟶1-1onto𝑇)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2147 {crab 3416 class class class wbr 5108 ↦ cmpt 5192 I cid 5552 ^◡ccnv 5657 ↾ cres 5660 ∘ ccom 5662 Fn wfn 6561 ⟶wf 6562 ⟶1-1wf1 6563 ⟶ontowfo 6564 ⟶1-1ontowf1o 6565 ‘cfv 6566 (class class class)co 7451 ↑_m cmap 8885 finSupp cfsupp 9391

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-supp 8215 df-1o 8514 df-map 8887 df-en 9012 df-dom 9013 df-fin 9015 df-fsupp 9392

This theorem is used by: mapfien2 9438 wemapwe 9735 oef1o 9736 fcobijfs 33409 fcobijfs2 33410

W3C validator