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Theorem psrbagconf1oOLD 21711

Description: Obsolete version of psrbagconf1o 21710 as of 6-Aug-2024. (Contributed by Mario Carneiro, 29-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
psrbag.d	⊢ 𝐷 = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
psrbagconf1o.s	⊢ 𝑆 = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_r ≤ 𝐹}

**Assertion**
Ref	Expression
psrbagconf1oOLD	⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → (𝑥 ∈ 𝑆 ↦ (𝐹 ∘_f − 𝑥)):𝑆–1-1-onto→𝑆)

Distinct variable groups: 𝑥,𝑓,𝑦,𝐹 𝑥,𝑉,𝑦 𝑓,𝐼,𝑥,𝑦 𝑥,𝑆 𝑥,𝐷,𝑦 Allowed substitution hints: 𝐷(𝑓) 𝑆(𝑦,𝑓) 𝑉(𝑓)

Proof of Theorem psrbagconf1oOLD Dummy variables 𝑛 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2730	. 2 ⊢ (𝑥 ∈ 𝑆 ↦ (𝐹 ∘_f − 𝑥)) = (𝑥 ∈ 𝑆 ↦ (𝐹 ∘_f − 𝑥))
2		simpll 763	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑥 ∈ 𝑆) → 𝐼 ∈ 𝑉)
3		simplr 765	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑥 ∈ 𝑆) → 𝐹 ∈ 𝐷)
4		simpr 483	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
5		breq1 5152	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦 ∘_r ≤ 𝐹 ↔ 𝑥 ∘_r ≤ 𝐹))
6		psrbagconf1o.s	. . . . . . . 8 ⊢ 𝑆 = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_r ≤ 𝐹}
7	5, 6	elrab2 3687	. . . . . . 7 ⊢ (𝑥 ∈ 𝑆 ↔ (𝑥 ∈ 𝐷 ∧ 𝑥 ∘_r ≤ 𝐹))
8	4, 7	sylib 217	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑥 ∈ 𝑆) → (𝑥 ∈ 𝐷 ∧ 𝑥 ∘_r ≤ 𝐹))
9	8	simpld 493	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐷)
10		psrbag.d	. . . . . 6 ⊢ 𝐷 = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
11	10	psrbagfOLD 21693	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ 𝐷) → 𝑥:𝐼⟶ℕ₀)
12	2, 9, 11	syl2anc 582	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑥 ∈ 𝑆) → 𝑥:𝐼⟶ℕ₀)
13	8	simprd 494	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∘_r ≤ 𝐹)
14	10	psrbagconOLD 21705	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝑥:𝐼⟶ℕ₀ ∧ 𝑥 ∘_r ≤ 𝐹)) → ((𝐹 ∘_f − 𝑥) ∈ 𝐷 ∧ (𝐹 ∘_f − 𝑥) ∘_r ≤ 𝐹))
15	2, 3, 12, 13, 14	syl13anc 1370	. . 3 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑥 ∈ 𝑆) → ((𝐹 ∘_f − 𝑥) ∈ 𝐷 ∧ (𝐹 ∘_f − 𝑥) ∘_r ≤ 𝐹))
16		breq1 5152	. . . 4 ⊢ (𝑦 = (𝐹 ∘_f − 𝑥) → (𝑦 ∘_r ≤ 𝐹 ↔ (𝐹 ∘_f − 𝑥) ∘_r ≤ 𝐹))
17	16, 6	elrab2 3687	. . 3 ⊢ ((𝐹 ∘_f − 𝑥) ∈ 𝑆 ↔ ((𝐹 ∘_f − 𝑥) ∈ 𝐷 ∧ (𝐹 ∘_f − 𝑥) ∘_r ≤ 𝐹))
18	15, 17	sylibr 233	. 2 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑥 ∈ 𝑆) → (𝐹 ∘_f − 𝑥) ∈ 𝑆)
19	18	ralrimiva 3144	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → ∀𝑥 ∈ 𝑆 (𝐹 ∘_f − 𝑥) ∈ 𝑆)
20		oveq2 7421	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝐹 ∘_f − 𝑥) = (𝐹 ∘_f − 𝑧))
21	20	eleq1d 2816	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝐹 ∘_f − 𝑥) ∈ 𝑆 ↔ (𝐹 ∘_f − 𝑧) ∈ 𝑆))
22	21	rspccva 3612	. . 3 ⊢ ((∀𝑥 ∈ 𝑆 (𝐹 ∘_f − 𝑥) ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) → (𝐹 ∘_f − 𝑧) ∈ 𝑆)
23	19, 22	sylan 578	. 2 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ 𝑧 ∈ 𝑆) → (𝐹 ∘_f − 𝑧) ∈ 𝑆)
24	10	psrbagfOLD 21693	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → 𝐹:𝐼⟶ℕ₀)
25	24	adantr 479	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝐹:𝐼⟶ℕ₀)
26	25	ffvelcdmda 7087	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) ∧ 𝑛 ∈ 𝐼) → (𝐹‘𝑛) ∈ ℕ₀)
27		simpll 763	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝐼 ∈ 𝑉)
28	6	ssrab3 4081	. . . . . . . . . 10 ⊢ 𝑆 ⊆ 𝐷
29		simprr 769	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑧 ∈ 𝑆)
30	28, 29	sselid 3981	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑧 ∈ 𝐷)
31	10	psrbagfOLD 21693	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝐷) → 𝑧:𝐼⟶ℕ₀)
32	27, 30, 31	syl2anc 582	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑧:𝐼⟶ℕ₀)
33	32	ffvelcdmda 7087	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) ∧ 𝑛 ∈ 𝐼) → (𝑧‘𝑛) ∈ ℕ₀)
34	12	adantrr 713	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑥:𝐼⟶ℕ₀)
35	34	ffvelcdmda 7087	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) ∧ 𝑛 ∈ 𝐼) → (𝑥‘𝑛) ∈ ℕ₀)
36		nn0cn 12488	. . . . . . . 8 ⊢ ((𝐹‘𝑛) ∈ ℕ₀ → (𝐹‘𝑛) ∈ ℂ)
37		nn0cn 12488	. . . . . . . 8 ⊢ ((𝑧‘𝑛) ∈ ℕ₀ → (𝑧‘𝑛) ∈ ℂ)
38		nn0cn 12488	. . . . . . . 8 ⊢ ((𝑥‘𝑛) ∈ ℕ₀ → (𝑥‘𝑛) ∈ ℂ)
39		subsub23 11471	. . . . . . . 8 ⊢ (((𝐹‘𝑛) ∈ ℂ ∧ (𝑧‘𝑛) ∈ ℂ ∧ (𝑥‘𝑛) ∈ ℂ) → (((𝐹‘𝑛) − (𝑧‘𝑛)) = (𝑥‘𝑛) ↔ ((𝐹‘𝑛) − (𝑥‘𝑛)) = (𝑧‘𝑛)))
40	36, 37, 38, 39	syl3an 1158	. . . . . . 7 ⊢ (((𝐹‘𝑛) ∈ ℕ₀ ∧ (𝑧‘𝑛) ∈ ℕ₀ ∧ (𝑥‘𝑛) ∈ ℕ₀) → (((𝐹‘𝑛) − (𝑧‘𝑛)) = (𝑥‘𝑛) ↔ ((𝐹‘𝑛) − (𝑥‘𝑛)) = (𝑧‘𝑛)))
41	26, 33, 35, 40	syl3anc 1369	. . . . . 6 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) ∧ 𝑛 ∈ 𝐼) → (((𝐹‘𝑛) − (𝑧‘𝑛)) = (𝑥‘𝑛) ↔ ((𝐹‘𝑛) − (𝑥‘𝑛)) = (𝑧‘𝑛)))
42		eqcom 2737	. . . . . 6 ⊢ ((𝑥‘𝑛) = ((𝐹‘𝑛) − (𝑧‘𝑛)) ↔ ((𝐹‘𝑛) − (𝑧‘𝑛)) = (𝑥‘𝑛))
43		eqcom 2737	. . . . . 6 ⊢ ((𝑧‘𝑛) = ((𝐹‘𝑛) − (𝑥‘𝑛)) ↔ ((𝐹‘𝑛) − (𝑥‘𝑛)) = (𝑧‘𝑛))
44	41, 42, 43	3bitr4g 313	. . . . 5 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) ∧ 𝑛 ∈ 𝐼) → ((𝑥‘𝑛) = ((𝐹‘𝑛) − (𝑧‘𝑛)) ↔ (𝑧‘𝑛) = ((𝐹‘𝑛) − (𝑥‘𝑛))))
45	25	ffnd 6719	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝐹 Fn 𝐼)
46	32	ffnd 6719	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑧 Fn 𝐼)
47		inidm 4219	. . . . . . 7 ⊢ (𝐼 ∩ 𝐼) = 𝐼
48		eqidd 2731	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) ∧ 𝑛 ∈ 𝐼) → (𝐹‘𝑛) = (𝐹‘𝑛))
49		eqidd 2731	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) ∧ 𝑛 ∈ 𝐼) → (𝑧‘𝑛) = (𝑧‘𝑛))
50	45, 46, 27, 27, 47, 48, 49	ofval 7685	. . . . . 6 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) ∧ 𝑛 ∈ 𝐼) → ((𝐹 ∘_f − 𝑧)‘𝑛) = ((𝐹‘𝑛) − (𝑧‘𝑛)))
51	50	eqeq2d 2741	. . . . 5 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) ∧ 𝑛 ∈ 𝐼) → ((𝑥‘𝑛) = ((𝐹 ∘_f − 𝑧)‘𝑛) ↔ (𝑥‘𝑛) = ((𝐹‘𝑛) − (𝑧‘𝑛))))
52	34	ffnd 6719	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → 𝑥 Fn 𝐼)
53		eqidd 2731	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) ∧ 𝑛 ∈ 𝐼) → (𝑥‘𝑛) = (𝑥‘𝑛))
54	45, 52, 27, 27, 47, 48, 53	ofval 7685	. . . . . 6 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) ∧ 𝑛 ∈ 𝐼) → ((𝐹 ∘_f − 𝑥)‘𝑛) = ((𝐹‘𝑛) − (𝑥‘𝑛)))
55	54	eqeq2d 2741	. . . . 5 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) ∧ 𝑛 ∈ 𝐼) → ((𝑧‘𝑛) = ((𝐹 ∘_f − 𝑥)‘𝑛) ↔ (𝑧‘𝑛) = ((𝐹‘𝑛) − (𝑥‘𝑛))))
56	44, 51, 55	3bitr4d 310	. . . 4 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) ∧ 𝑛 ∈ 𝐼) → ((𝑥‘𝑛) = ((𝐹 ∘_f − 𝑧)‘𝑛) ↔ (𝑧‘𝑛) = ((𝐹 ∘_f − 𝑥)‘𝑛)))
57	56	ralbidva 3173	. . 3 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (∀𝑛 ∈ 𝐼 (𝑥‘𝑛) = ((𝐹 ∘_f − 𝑧)‘𝑛) ↔ ∀𝑛 ∈ 𝐼 (𝑧‘𝑛) = ((𝐹 ∘_f − 𝑥)‘𝑛)))
58	23	adantrl 712	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝐹 ∘_f − 𝑧) ∈ 𝑆)
59	28, 58	sselid 3981	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝐹 ∘_f − 𝑧) ∈ 𝐷)
60	10	psrbagfOLD 21693	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∘_f − 𝑧) ∈ 𝐷) → (𝐹 ∘_f − 𝑧):𝐼⟶ℕ₀)
61	27, 59, 60	syl2anc 582	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝐹 ∘_f − 𝑧):𝐼⟶ℕ₀)
62	61	ffnd 6719	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝐹 ∘_f − 𝑧) Fn 𝐼)
63		eqfnfv 7033	. . . 4 ⊢ ((𝑥 Fn 𝐼 ∧ (𝐹 ∘_f − 𝑧) Fn 𝐼) → (𝑥 = (𝐹 ∘_f − 𝑧) ↔ ∀𝑛 ∈ 𝐼 (𝑥‘𝑛) = ((𝐹 ∘_f − 𝑧)‘𝑛)))
64	52, 62, 63	syl2anc 582	. . 3 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥 = (𝐹 ∘_f − 𝑧) ↔ ∀𝑛 ∈ 𝐼 (𝑥‘𝑛) = ((𝐹 ∘_f − 𝑧)‘𝑛)))
65	18	adantrr 713	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝐹 ∘_f − 𝑥) ∈ 𝑆)
66	28, 65	sselid 3981	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝐹 ∘_f − 𝑥) ∈ 𝐷)
67	10	psrbagfOLD 21693	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∘_f − 𝑥) ∈ 𝐷) → (𝐹 ∘_f − 𝑥):𝐼⟶ℕ₀)
68	27, 66, 67	syl2anc 582	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝐹 ∘_f − 𝑥):𝐼⟶ℕ₀)
69	68	ffnd 6719	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝐹 ∘_f − 𝑥) Fn 𝐼)
70		eqfnfv 7033	. . . 4 ⊢ ((𝑧 Fn 𝐼 ∧ (𝐹 ∘_f − 𝑥) Fn 𝐼) → (𝑧 = (𝐹 ∘_f − 𝑥) ↔ ∀𝑛 ∈ 𝐼 (𝑧‘𝑛) = ((𝐹 ∘_f − 𝑥)‘𝑛)))
71	46, 69, 70	syl2anc 582	. . 3 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑧 = (𝐹 ∘_f − 𝑥) ↔ ∀𝑛 ∈ 𝐼 (𝑧‘𝑛) = ((𝐹 ∘_f − 𝑥)‘𝑛)))
72	57, 64, 71	3bitr4d 310	. 2 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) ∧ (𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥 = (𝐹 ∘_f − 𝑧) ↔ 𝑧 = (𝐹 ∘_f − 𝑥)))
73	1, 18, 23, 72	f1o2d 7664	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷) → (𝑥 ∈ 𝑆 ↦ (𝐹 ∘_f − 𝑥)):𝑆–1-1-onto→𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ∀wral 3059 {crab 3430 class class class wbr 5149 ↦ cmpt 5232 ^◡ccnv 5676 “ cima 5680 Fn wfn 6539 ⟶wf 6540 –1-1-onto→wf1o 6543 ‘cfv 6544 (class class class)co 7413 ∘_f cof 7672 ∘_r cofr 7673 ↑_m cmap 8824 Fincfn 8943 ℂcc 11112 ≤ cle 11255 − cmin 11450 ℕcn 12218 ℕ₀cn0 12478

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7674 df-ofr 7675 df-om 7860 df-2nd 7980 df-supp 8151 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-nn 12219 df-n0 12479

This theorem is referenced by: psrass1lemOLD 21714

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