f1cof1b - Mathbox for Alexander van der Vekens

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Theorem f1cof1b 47703

Description: If the range of 𝐹 equals the domain of 𝐺, then the composition (𝐺 ∘ 𝐹) is injective iff 𝐹 and 𝐺 are both injective. (Contributed by GL and AV, 19-Sep-2024.)

**Assertion**
Ref	Expression
f1cof1b	⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–1-1→𝐷 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷)))

**Proof of Theorem f1cof1b**
Step	Hyp	Ref	Expression
1		simp1 1152	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → 𝐹:𝐴⟶𝐵)
2		eqid 2769	. . . . . . . . 9 ⊢ (ran 𝐹 ∩ 𝐶) = (ran 𝐹 ∩ 𝐶)
3		eqid 2769	. . . . . . . . 9 ⊢ (^◡𝐹 “ 𝐶) = (^◡𝐹 “ 𝐶)
4		eqid 2769	. . . . . . . . 9 ⊢ (𝐹 ↾ (^◡𝐹 “ 𝐶)) = (𝐹 ↾ (^◡𝐹 “ 𝐶))
5		simp2 1153	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → 𝐺:𝐶⟶𝐷)
6		eqid 2769	. . . . . . . . 9 ⊢ (𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = (𝐺 ↾ (ran 𝐹 ∩ 𝐶))
7		simp3 1154	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ran 𝐹 = 𝐶)
8	1, 2, 3, 4, 5, 6, 7	f1cof1blem 47700	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → (((^◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐹 ↾ (^◡𝐹 “ 𝐶)) = 𝐹 ∧ (𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺)))
9		simpll 778	. . . . . . . 8 ⊢ ((((^◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐹 ↾ (^◡𝐹 “ 𝐶)) = 𝐹 ∧ (𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺)) → (^◡𝐹 “ 𝐶) = 𝐴)
10		f1eq2 6771	. . . . . . . 8 ⊢ ((^◡𝐹 “ 𝐶) = 𝐴 → ((𝐺 ∘ 𝐹):(^◡𝐹 “ 𝐶)–1-1→𝐷 ↔ (𝐺 ∘ 𝐹):𝐴–1-1→𝐷))
11	8, 9, 10	3syl 19	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):(^◡𝐹 “ 𝐶)–1-1→𝐷 ↔ (𝐺 ∘ 𝐹):𝐴–1-1→𝐷))
12	11	bicomd 226	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–1-1→𝐷 ↔ (𝐺 ∘ 𝐹):(^◡𝐹 “ 𝐶)–1-1→𝐷))
13		ancom 465	. . . . . . . . . 10 ⊢ (((𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺 ∧ (𝐹 ↾ (^◡𝐹 “ 𝐶)) = 𝐹) ↔ ((𝐹 ↾ (^◡𝐹 “ 𝐶)) = 𝐹 ∧ (𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺))
14	13	anbi2i 634	. . . . . . . . 9 ⊢ ((((^◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺 ∧ (𝐹 ↾ (^◡𝐹 “ 𝐶)) = 𝐹)) ↔ (((^◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐹 ↾ (^◡𝐹 “ 𝐶)) = 𝐹 ∧ (𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺)))
15	8, 14	sylibr 237	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → (((^◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺 ∧ (𝐹 ↾ (^◡𝐹 “ 𝐶)) = 𝐹)))
16	15	adantr 485	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺 ∘ 𝐹):(^◡𝐹 “ 𝐶)–1-1→𝐷) → (((^◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺 ∧ (𝐹 ↾ (^◡𝐹 “ 𝐶)) = 𝐹)))
17	1	adantr 485	. . . . . . . . 9 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺 ∘ 𝐹):(^◡𝐹 “ 𝐶)–1-1→𝐷) → 𝐹:𝐴⟶𝐵)
18	5	adantr 485	. . . . . . . . 9 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺 ∘ 𝐹):(^◡𝐹 “ 𝐶)–1-1→𝐷) → 𝐺:𝐶⟶𝐷)
19		simpr 489	. . . . . . . . 9 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺 ∘ 𝐹):(^◡𝐹 “ 𝐶)–1-1→𝐷) → (𝐺 ∘ 𝐹):(^◡𝐹 “ 𝐶)–1-1→𝐷)
20	17, 2, 3, 4, 18, 6, 19	fcoresf1 47695	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺 ∘ 𝐹):(^◡𝐹 “ 𝐶)–1-1→𝐷) → ((𝐹 ↾ (^◡𝐹 “ 𝐶)):(^◡𝐹 “ 𝐶)–1-1→(ran 𝐹 ∩ 𝐶) ∧ (𝐺 ↾ (ran 𝐹 ∩ 𝐶)):(ran 𝐹 ∩ 𝐶)–1-1→𝐷))
21	20	ancomd 466	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺 ∘ 𝐹):(^◡𝐹 “ 𝐶)–1-1→𝐷) → ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)):(ran 𝐹 ∩ 𝐶)–1-1→𝐷 ∧ (𝐹 ↾ (^◡𝐹 “ 𝐶)):(^◡𝐹 “ 𝐶)–1-1→(ran 𝐹 ∩ 𝐶)))
22		simprl 782	. . . . . . . . . 10 ⊢ ((((^◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺 ∧ (𝐹 ↾ (^◡𝐹 “ 𝐶)) = 𝐹)) → (𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺)
23		simpr 489	. . . . . . . . . . 11 ⊢ (((^◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) → (ran 𝐹 ∩ 𝐶) = 𝐶)
24	23	adantr 485	. . . . . . . . . 10 ⊢ ((((^◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺 ∧ (𝐹 ↾ (^◡𝐹 “ 𝐶)) = 𝐹)) → (ran 𝐹 ∩ 𝐶) = 𝐶)
25		eqidd 2770	. . . . . . . . . 10 ⊢ ((((^◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺 ∧ (𝐹 ↾ (^◡𝐹 “ 𝐶)) = 𝐹)) → 𝐷 = 𝐷)
26	22, 24, 25	f1eq123d 6813	. . . . . . . . 9 ⊢ ((((^◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺 ∧ (𝐹 ↾ (^◡𝐹 “ 𝐶)) = 𝐹)) → ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)):(ran 𝐹 ∩ 𝐶)–1-1→𝐷 ↔ 𝐺:𝐶–1-1→𝐷))
27	26	biimpd 232	. . . . . . . 8 ⊢ ((((^◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺 ∧ (𝐹 ↾ (^◡𝐹 “ 𝐶)) = 𝐹)) → ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)):(ran 𝐹 ∩ 𝐶)–1-1→𝐷 → 𝐺:𝐶–1-1→𝐷))
28		simprr 784	. . . . . . . . . 10 ⊢ ((((^◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺 ∧ (𝐹 ↾ (^◡𝐹 “ 𝐶)) = 𝐹)) → (𝐹 ↾ (^◡𝐹 “ 𝐶)) = 𝐹)
29		simpll 778	. . . . . . . . . 10 ⊢ ((((^◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺 ∧ (𝐹 ↾ (^◡𝐹 “ 𝐶)) = 𝐹)) → (^◡𝐹 “ 𝐶) = 𝐴)
30	28, 29, 24	f1eq123d 6813	. . . . . . . . 9 ⊢ ((((^◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺 ∧ (𝐹 ↾ (^◡𝐹 “ 𝐶)) = 𝐹)) → ((𝐹 ↾ (^◡𝐹 “ 𝐶)):(^◡𝐹 “ 𝐶)–1-1→(ran 𝐹 ∩ 𝐶) ↔ 𝐹:𝐴–1-1→𝐶))
31	30	biimpd 232	. . . . . . . 8 ⊢ ((((^◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺 ∧ (𝐹 ↾ (^◡𝐹 “ 𝐶)) = 𝐹)) → ((𝐹 ↾ (^◡𝐹 “ 𝐶)):(^◡𝐹 “ 𝐶)–1-1→(ran 𝐹 ∩ 𝐶) → 𝐹:𝐴–1-1→𝐶))
32	27, 31	anim12d 620	. . . . . . 7 ⊢ ((((^◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺 ∧ (𝐹 ↾ (^◡𝐹 “ 𝐶)) = 𝐹)) → (((𝐺 ↾ (ran 𝐹 ∩ 𝐶)):(ran 𝐹 ∩ 𝐶)–1-1→𝐷 ∧ (𝐹 ↾ (^◡𝐹 “ 𝐶)):(^◡𝐹 “ 𝐶)–1-1→(ran 𝐹 ∩ 𝐶)) → (𝐺:𝐶–1-1→𝐷 ∧ 𝐹:𝐴–1-1→𝐶)))
33	16, 21, 32	sylc 66	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺 ∘ 𝐹):(^◡𝐹 “ 𝐶)–1-1→𝐷) → (𝐺:𝐶–1-1→𝐷 ∧ 𝐹:𝐴–1-1→𝐶))
34	12, 33	sylbida 603	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺 ∘ 𝐹):𝐴–1-1→𝐷) → (𝐺:𝐶–1-1→𝐷 ∧ 𝐹:𝐴–1-1→𝐶))
35		ffrn 6720	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶ran 𝐹)
36		ax-1 6	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹:𝐴⟶ran 𝐹 → 𝐹:𝐴⟶𝐵))
37	35, 36	impbid2 229	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶ran 𝐹))
38	37	anbi1d 642	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐹:𝐴⟶𝐵 ∧ Fun ^◡𝐹) ↔ (𝐹:𝐴⟶ran 𝐹 ∧ Fun ^◡𝐹)))
39		df-f1 6542	. . . . . . . . . 10 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ^◡𝐹))
40		df-f1 6542	. . . . . . . . . 10 ⊢ (𝐹:𝐴–1-1→ran 𝐹 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ Fun ^◡𝐹))
41	38, 39, 40	3bitr4g 317	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:𝐴–1-1→ran 𝐹))
42	41	3ad2ant1 1149	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:𝐴–1-1→ran 𝐹))
43		f1eq3 6772	. . . . . . . . 9 ⊢ (ran 𝐹 = 𝐶 → (𝐹:𝐴–1-1→ran 𝐹 ↔ 𝐹:𝐴–1-1→𝐶))
44	43	3ad2ant3 1151	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → (𝐹:𝐴–1-1→ran 𝐹 ↔ 𝐹:𝐴–1-1→𝐶))
45	42, 44	bitrd 282	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:𝐴–1-1→𝐶))
46	45	anbi2d 641	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺:𝐶–1-1→𝐷 ∧ 𝐹:𝐴–1-1→𝐵) ↔ (𝐺:𝐶–1-1→𝐷 ∧ 𝐹:𝐴–1-1→𝐶)))
47	46	adantr 485	. . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺 ∘ 𝐹):𝐴–1-1→𝐷) → ((𝐺:𝐶–1-1→𝐷 ∧ 𝐹:𝐴–1-1→𝐵) ↔ (𝐺:𝐶–1-1→𝐷 ∧ 𝐹:𝐴–1-1→𝐶)))
48	34, 47	mpbird 260	. . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺 ∘ 𝐹):𝐴–1-1→𝐷) → (𝐺:𝐶–1-1→𝐷 ∧ 𝐹:𝐴–1-1→𝐵))
49	48	ancomd 466	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺 ∘ 𝐹):𝐴–1-1→𝐷) → (𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷))
50	49	ex 417	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–1-1→𝐷 → (𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷)))
51		f1cof1 6787	. . . 4 ⊢ ((𝐺:𝐶–1-1→𝐷 ∧ 𝐹:𝐴–1-1→𝐵) → (𝐺 ∘ 𝐹):(^◡𝐹 “ 𝐶)–1-1→𝐷)
52	51	ancoms 463	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷) → (𝐺 ∘ 𝐹):(^◡𝐹 “ 𝐶)–1-1→𝐷)
53		imaeq2 6059	. . . . . . . 8 ⊢ (𝐶 = ran 𝐹 → (^◡𝐹 “ 𝐶) = (^◡𝐹 “ ran 𝐹))
54		cnvimarndm 6086	. . . . . . . 8 ⊢ (^◡𝐹 “ ran 𝐹) = dom 𝐹
55	53, 54	eqtrdi 2820	. . . . . . 7 ⊢ (𝐶 = ran 𝐹 → (^◡𝐹 “ 𝐶) = dom 𝐹)
56	55	eqcoms 2777	. . . . . 6 ⊢ (ran 𝐹 = 𝐶 → (^◡𝐹 “ 𝐶) = dom 𝐹)
57	56	3ad2ant3 1151	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → (^◡𝐹 “ 𝐶) = dom 𝐹)
58	1	fdmd 6717	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → dom 𝐹 = 𝐴)
59	57, 58	eqtrd 2804	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → (^◡𝐹 “ 𝐶) = 𝐴)
60	59, 10	syl 18	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):(^◡𝐹 “ 𝐶)–1-1→𝐷 ↔ (𝐺 ∘ 𝐹):𝐴–1-1→𝐷))
61	52, 60	imbitrid 247	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷) → (𝐺 ∘ 𝐹):𝐴–1-1→𝐷))
62	50, 61	impbid 215	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–1-1→𝐷 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1101 = wceq 1567 ∩ cin 3912 ^◡ccnv 5661 dom cdm 5662 ran crn 5663 ↾ cres 5664 “ cima 5665 ∘ ccom 5666 Fun wfun 6531 ⟶wf 6533 –1-1→wf1 6534

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5261 ax-nul 5271 ax-pr 5405

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-ral 3086 df-rex 3096 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4295 df-if 4493 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-br 5114 df-opab 5178 df-mpt 5197 df-id 5557 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-fv 6545

This theorem is referenced by: f1ocof1ob 47707

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