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Theorem f1cof1b 46083
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⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) β†’ ((𝐺 ∘ 𝐹):𝐴–1-1→𝐷 ↔ (𝐹:𝐴–1-1→𝐡 ∧ 𝐺:𝐢–1-1→𝐷)))
Proof of Theorem f1cof1b
StepHypRef Expression
1 simp1 1134 . . . . . . . . 9 ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) β†’ 𝐹:𝐴⟢𝐡)
2 eqid 2730 . . . . . . . . 9 (ran 𝐹 ∩ 𝐢) = (ran 𝐹 ∩ 𝐢)
3 eqid 2730 . . . . . . . . 9 (◑𝐹 β€œ 𝐢) = (◑𝐹 β€œ 𝐢)
4 eqid 2730 . . . . . . . . 9 (𝐹 β†Ύ (◑𝐹 β€œ 𝐢)) = (𝐹 β†Ύ (◑𝐹 β€œ 𝐢))
5 simp2 1135 . . . . . . . . 9 ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) β†’ 𝐺:𝐢⟢𝐷)
6 eqid 2730 . . . . . . . . 9 (𝐺 β†Ύ (ran 𝐹 ∩ 𝐢)) = (𝐺 β†Ύ (ran 𝐹 ∩ 𝐢))
7 simp3 1136 . . . . . . . . 9 ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) β†’ ran 𝐹 = 𝐢)
81, 2, 3, 4, 5, 6, 7f1cof1blem 46082 . . . . . . . 8 ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) β†’ (((◑𝐹 β€œ 𝐢) = 𝐴 ∧ (ran 𝐹 ∩ 𝐢) = 𝐢) ∧ ((𝐹 β†Ύ (◑𝐹 β€œ 𝐢)) = 𝐹 ∧ (𝐺 β†Ύ (ran 𝐹 ∩ 𝐢)) = 𝐺)))
9 simpll 763 . . . . . . . 8 ((((◑𝐹 β€œ 𝐢) = 𝐴 ∧ (ran 𝐹 ∩ 𝐢) = 𝐢) ∧ ((𝐹 β†Ύ (◑𝐹 β€œ 𝐢)) = 𝐹 ∧ (𝐺 β†Ύ (ran 𝐹 ∩ 𝐢)) = 𝐺)) β†’ (◑𝐹 β€œ 𝐢) = 𝐴)
10 f1eq2 6782 . . . . . . . 8 ((◑𝐹 β€œ 𝐢) = 𝐴 β†’ ((𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐢)–1-1→𝐷 ↔ (𝐺 ∘ 𝐹):𝐴–1-1→𝐷))
118, 9, 103syl 18 . . . . . . 7 ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) β†’ ((𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐢)–1-1→𝐷 ↔ (𝐺 ∘ 𝐹):𝐴–1-1→𝐷))
1211bicomd 222 . . . . . 6 ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) β†’ ((𝐺 ∘ 𝐹):𝐴–1-1→𝐷 ↔ (𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐢)–1-1→𝐷))
13 ancom 459 . . . . . . . . . 10 (((𝐺 β†Ύ (ran 𝐹 ∩ 𝐢)) = 𝐺 ∧ (𝐹 β†Ύ (◑𝐹 β€œ 𝐢)) = 𝐹) ↔ ((𝐹 β†Ύ (◑𝐹 β€œ 𝐢)) = 𝐹 ∧ (𝐺 β†Ύ (ran 𝐹 ∩ 𝐢)) = 𝐺))
1413anbi2i 621 . . . . . . . . 9 ((((◑𝐹 β€œ 𝐢) = 𝐴 ∧ (ran 𝐹 ∩ 𝐢) = 𝐢) ∧ ((𝐺 β†Ύ (ran 𝐹 ∩ 𝐢)) = 𝐺 ∧ (𝐹 β†Ύ (◑𝐹 β€œ 𝐢)) = 𝐹)) ↔ (((◑𝐹 β€œ 𝐢) = 𝐴 ∧ (ran 𝐹 ∩ 𝐢) = 𝐢) ∧ ((𝐹 β†Ύ (◑𝐹 β€œ 𝐢)) = 𝐹 ∧ (𝐺 β†Ύ (ran 𝐹 ∩ 𝐢)) = 𝐺)))
158, 14sylibr 233 . . . . . . . 8 ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) β†’ (((◑𝐹 β€œ 𝐢) = 𝐴 ∧ (ran 𝐹 ∩ 𝐢) = 𝐢) ∧ ((𝐺 β†Ύ (ran 𝐹 ∩ 𝐢)) = 𝐺 ∧ (𝐹 β†Ύ (◑𝐹 β€œ 𝐢)) = 𝐹)))
1615adantr 479 . . . . . . 7 (((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) ∧ (𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐢)–1-1→𝐷) β†’ (((◑𝐹 β€œ 𝐢) = 𝐴 ∧ (ran 𝐹 ∩ 𝐢) = 𝐢) ∧ ((𝐺 β†Ύ (ran 𝐹 ∩ 𝐢)) = 𝐺 ∧ (𝐹 β†Ύ (◑𝐹 β€œ 𝐢)) = 𝐹)))
171adantr 479 . . . . . . . . 9 (((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) ∧ (𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐢)–1-1→𝐷) β†’ 𝐹:𝐴⟢𝐡)
185adantr 479 . . . . . . . . 9 (((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) ∧ (𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐢)–1-1→𝐷) β†’ 𝐺:𝐢⟢𝐷)
19 simpr 483 . . . . . . . . 9 (((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) ∧ (𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐢)–1-1→𝐷) β†’ (𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐢)–1-1→𝐷)
2017, 2, 3, 4, 18, 6, 19fcoresf1 46077 . . . . . . . 8 (((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) ∧ (𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐢)–1-1→𝐷) β†’ ((𝐹 β†Ύ (◑𝐹 β€œ 𝐢)):(◑𝐹 β€œ 𝐢)–1-1β†’(ran 𝐹 ∩ 𝐢) ∧ (𝐺 β†Ύ (ran 𝐹 ∩ 𝐢)):(ran 𝐹 ∩ 𝐢)–1-1→𝐷))
2120ancomd 460 . . . . . . 7 (((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) ∧ (𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐢)–1-1→𝐷) β†’ ((𝐺 β†Ύ (ran 𝐹 ∩ 𝐢)):(ran 𝐹 ∩ 𝐢)–1-1→𝐷 ∧ (𝐹 β†Ύ (◑𝐹 β€œ 𝐢)):(◑𝐹 β€œ 𝐢)–1-1β†’(ran 𝐹 ∩ 𝐢)))
22 simprl 767 . . . . . . . . . 10 ((((◑𝐹 β€œ 𝐢) = 𝐴 ∧ (ran 𝐹 ∩ 𝐢) = 𝐢) ∧ ((𝐺 β†Ύ (ran 𝐹 ∩ 𝐢)) = 𝐺 ∧ (𝐹 β†Ύ (◑𝐹 β€œ 𝐢)) = 𝐹)) β†’ (𝐺 β†Ύ (ran 𝐹 ∩ 𝐢)) = 𝐺)
23 simpr 483 . . . . . . . . . . 11 (((◑𝐹 β€œ 𝐢) = 𝐴 ∧ (ran 𝐹 ∩ 𝐢) = 𝐢) β†’ (ran 𝐹 ∩ 𝐢) = 𝐢)
2423adantr 479 . . . . . . . . . 10 ((((◑𝐹 β€œ 𝐢) = 𝐴 ∧ (ran 𝐹 ∩ 𝐢) = 𝐢) ∧ ((𝐺 β†Ύ (ran 𝐹 ∩ 𝐢)) = 𝐺 ∧ (𝐹 β†Ύ (◑𝐹 β€œ 𝐢)) = 𝐹)) β†’ (ran 𝐹 ∩ 𝐢) = 𝐢)
25 eqidd 2731 . . . . . . . . . 10 ((((◑𝐹 β€œ 𝐢) = 𝐴 ∧ (ran 𝐹 ∩ 𝐢) = 𝐢) ∧ ((𝐺 β†Ύ (ran 𝐹 ∩ 𝐢)) = 𝐺 ∧ (𝐹 β†Ύ (◑𝐹 β€œ 𝐢)) = 𝐹)) β†’ 𝐷 = 𝐷)
2622, 24, 25f1eq123d 6824 . . . . . . . . 9 ((((◑𝐹 β€œ 𝐢) = 𝐴 ∧ (ran 𝐹 ∩ 𝐢) = 𝐢) ∧ ((𝐺 β†Ύ (ran 𝐹 ∩ 𝐢)) = 𝐺 ∧ (𝐹 β†Ύ (◑𝐹 β€œ 𝐢)) = 𝐹)) β†’ ((𝐺 β†Ύ (ran 𝐹 ∩ 𝐢)):(ran 𝐹 ∩ 𝐢)–1-1→𝐷 ↔ 𝐺:𝐢–1-1→𝐷))
2726biimpd 228 . . . . . . . 8 ((((◑𝐹 β€œ 𝐢) = 𝐴 ∧ (ran 𝐹 ∩ 𝐢) = 𝐢) ∧ ((𝐺 β†Ύ (ran 𝐹 ∩ 𝐢)) = 𝐺 ∧ (𝐹 β†Ύ (◑𝐹 β€œ 𝐢)) = 𝐹)) β†’ ((𝐺 β†Ύ (ran 𝐹 ∩ 𝐢)):(ran 𝐹 ∩ 𝐢)–1-1→𝐷 β†’ 𝐺:𝐢–1-1→𝐷))
28 simprr 769 . . . . . . . . . 10 ((((◑𝐹 β€œ 𝐢) = 𝐴 ∧ (ran 𝐹 ∩ 𝐢) = 𝐢) ∧ ((𝐺 β†Ύ (ran 𝐹 ∩ 𝐢)) = 𝐺 ∧ (𝐹 β†Ύ (◑𝐹 β€œ 𝐢)) = 𝐹)) β†’ (𝐹 β†Ύ (◑𝐹 β€œ 𝐢)) = 𝐹)
29 simpll 763 . . . . . . . . . 10 ((((◑𝐹 β€œ 𝐢) = 𝐴 ∧ (ran 𝐹 ∩ 𝐢) = 𝐢) ∧ ((𝐺 β†Ύ (ran 𝐹 ∩ 𝐢)) = 𝐺 ∧ (𝐹 β†Ύ (◑𝐹 β€œ 𝐢)) = 𝐹)) β†’ (◑𝐹 β€œ 𝐢) = 𝐴)
3028, 29, 24f1eq123d 6824 . . . . . . . . 9 ((((◑𝐹 β€œ 𝐢) = 𝐴 ∧ (ran 𝐹 ∩ 𝐢) = 𝐢) ∧ ((𝐺 β†Ύ (ran 𝐹 ∩ 𝐢)) = 𝐺 ∧ (𝐹 β†Ύ (◑𝐹 β€œ 𝐢)) = 𝐹)) β†’ ((𝐹 β†Ύ (◑𝐹 β€œ 𝐢)):(◑𝐹 β€œ 𝐢)–1-1β†’(ran 𝐹 ∩ 𝐢) ↔ 𝐹:𝐴–1-1→𝐢))
3130biimpd 228 . . . . . . . 8 ((((◑𝐹 β€œ 𝐢) = 𝐴 ∧ (ran 𝐹 ∩ 𝐢) = 𝐢) ∧ ((𝐺 β†Ύ (ran 𝐹 ∩ 𝐢)) = 𝐺 ∧ (𝐹 β†Ύ (◑𝐹 β€œ 𝐢)) = 𝐹)) β†’ ((𝐹 β†Ύ (◑𝐹 β€œ 𝐢)):(◑𝐹 β€œ 𝐢)–1-1β†’(ran 𝐹 ∩ 𝐢) β†’ 𝐹:𝐴–1-1→𝐢))
3227, 31anim12d 607 . . . . . . 7 ((((◑𝐹 β€œ 𝐢) = 𝐴 ∧ (ran 𝐹 ∩ 𝐢) = 𝐢) ∧ ((𝐺 β†Ύ (ran 𝐹 ∩ 𝐢)) = 𝐺 ∧ (𝐹 β†Ύ (◑𝐹 β€œ 𝐢)) = 𝐹)) β†’ (((𝐺 β†Ύ (ran 𝐹 ∩ 𝐢)):(ran 𝐹 ∩ 𝐢)–1-1→𝐷 ∧ (𝐹 β†Ύ (◑𝐹 β€œ 𝐢)):(◑𝐹 β€œ 𝐢)–1-1β†’(ran 𝐹 ∩ 𝐢)) β†’ (𝐺:𝐢–1-1→𝐷 ∧ 𝐹:𝐴–1-1→𝐢)))
3316, 21, 32sylc 65 . . . . . 6 (((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) ∧ (𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐢)–1-1→𝐷) β†’ (𝐺:𝐢–1-1→𝐷 ∧ 𝐹:𝐴–1-1→𝐢))
3412, 33sylbida 590 . . . . 5 (((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) ∧ (𝐺 ∘ 𝐹):𝐴–1-1→𝐷) β†’ (𝐺:𝐢–1-1→𝐷 ∧ 𝐹:𝐴–1-1→𝐢))
35 ffrn 6730 . . . . . . . . . . . 12 (𝐹:𝐴⟢𝐡 β†’ 𝐹:𝐴⟢ran 𝐹)
36 ax-1 6 . . . . . . . . . . . 12 (𝐹:𝐴⟢𝐡 β†’ (𝐹:𝐴⟢ran 𝐹 β†’ 𝐹:𝐴⟢𝐡))
3735, 36impbid2 225 . . . . . . . . . . 11 (𝐹:𝐴⟢𝐡 β†’ (𝐹:𝐴⟢𝐡 ↔ 𝐹:𝐴⟢ran 𝐹))
3837anbi1d 628 . . . . . . . . . 10 (𝐹:𝐴⟢𝐡 β†’ ((𝐹:𝐴⟢𝐡 ∧ Fun ◑𝐹) ↔ (𝐹:𝐴⟢ran 𝐹 ∧ Fun ◑𝐹)))
39 df-f1 6547 . . . . . . . . . 10 (𝐹:𝐴–1-1→𝐡 ↔ (𝐹:𝐴⟢𝐡 ∧ Fun ◑𝐹))
40 df-f1 6547 . . . . . . . . . 10 (𝐹:𝐴–1-1β†’ran 𝐹 ↔ (𝐹:𝐴⟢ran 𝐹 ∧ Fun ◑𝐹))
4138, 39, 403bitr4g 313 . . . . . . . . 9 (𝐹:𝐴⟢𝐡 β†’ (𝐹:𝐴–1-1→𝐡 ↔ 𝐹:𝐴–1-1β†’ran 𝐹))
42413ad2ant1 1131 . . . . . . . 8 ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) β†’ (𝐹:𝐴–1-1→𝐡 ↔ 𝐹:𝐴–1-1β†’ran 𝐹))
43 f1eq3 6783 . . . . . . . . 9 (ran 𝐹 = 𝐢 β†’ (𝐹:𝐴–1-1β†’ran 𝐹 ↔ 𝐹:𝐴–1-1→𝐢))
44433ad2ant3 1133 . . . . . . . 8 ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) β†’ (𝐹:𝐴–1-1β†’ran 𝐹 ↔ 𝐹:𝐴–1-1→𝐢))
4542, 44bitrd 278 . . . . . . 7 ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) β†’ (𝐹:𝐴–1-1→𝐡 ↔ 𝐹:𝐴–1-1→𝐢))
4645anbi2d 627 . . . . . 6 ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) β†’ ((𝐺:𝐢–1-1→𝐷 ∧ 𝐹:𝐴–1-1→𝐡) ↔ (𝐺:𝐢–1-1→𝐷 ∧ 𝐹:𝐴–1-1→𝐢)))
4746adantr 479 . . . . 5 (((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) ∧ (𝐺 ∘ 𝐹):𝐴–1-1→𝐷) β†’ ((𝐺:𝐢–1-1→𝐷 ∧ 𝐹:𝐴–1-1→𝐡) ↔ (𝐺:𝐢–1-1→𝐷 ∧ 𝐹:𝐴–1-1→𝐢)))
4834, 47mpbird 256 . . . 4 (((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) ∧ (𝐺 ∘ 𝐹):𝐴–1-1→𝐷) β†’ (𝐺:𝐢–1-1→𝐷 ∧ 𝐹:𝐴–1-1→𝐡))
4948ancomd 460 . . 3 (((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) ∧ (𝐺 ∘ 𝐹):𝐴–1-1→𝐷) β†’ (𝐹:𝐴–1-1→𝐡 ∧ 𝐺:𝐢–1-1→𝐷))
5049ex 411 . 2 ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) β†’ ((𝐺 ∘ 𝐹):𝐴–1-1→𝐷 β†’ (𝐹:𝐴–1-1→𝐡 ∧ 𝐺:𝐢–1-1→𝐷)))
51 f1cof1 6797 . . . 4 ((𝐺:𝐢–1-1→𝐷 ∧ 𝐹:𝐴–1-1→𝐡) β†’ (𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐢)–1-1→𝐷)
5251ancoms 457 . . 3 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐺:𝐢–1-1→𝐷) β†’ (𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐢)–1-1→𝐷)
53 imaeq2 6054 . . . . . . . 8 (𝐢 = ran 𝐹 β†’ (◑𝐹 β€œ 𝐢) = (◑𝐹 β€œ ran 𝐹))
54 cnvimarndm 6080 . . . . . . . 8 (◑𝐹 β€œ ran 𝐹) = dom 𝐹
5553, 54eqtrdi 2786 . . . . . . 7 (𝐢 = ran 𝐹 β†’ (◑𝐹 β€œ 𝐢) = dom 𝐹)
5655eqcoms 2738 . . . . . 6 (ran 𝐹 = 𝐢 β†’ (◑𝐹 β€œ 𝐢) = dom 𝐹)
57563ad2ant3 1133 . . . . 5 ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) β†’ (◑𝐹 β€œ 𝐢) = dom 𝐹)
581fdmd 6727 . . . . 5 ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) β†’ dom 𝐹 = 𝐴)
5957, 58eqtrd 2770 . . . 4 ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) β†’ (◑𝐹 β€œ 𝐢) = 𝐴)
6059, 10syl 17 . . 3 ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) β†’ ((𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐢)–1-1→𝐷 ↔ (𝐺 ∘ 𝐹):𝐴–1-1→𝐷))
6152, 60imbitrid 243 . 2 ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) β†’ ((𝐹:𝐴–1-1→𝐡 ∧ 𝐺:𝐢–1-1→𝐷) β†’ (𝐺 ∘ 𝐹):𝐴–1-1→𝐷))
6250, 61impbid 211 1 ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) β†’ ((𝐺 ∘ 𝐹):𝐴–1-1→𝐷 ↔ (𝐹:𝐴–1-1→𝐡 ∧ 𝐺:𝐢–1-1→𝐷)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∩ cin 3946  β—‘ccnv 5674  dom cdm 5675  ran crn 5676   β†Ύ cres 5677   β€œ cima 5678   ∘ ccom 5679  Fun wfun 6536  βŸΆwf 6538  β€“1-1β†’wf1 6539
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-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  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-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-fv 6550
This theorem is referenced by:  f1ocof1ob  46087
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