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Theorem f1cof1b 45785
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 1137 . . . . . . . . 9 ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) β†’ 𝐹:𝐴⟢𝐡)
2 eqid 2733 . . . . . . . . 9 (ran 𝐹 ∩ 𝐢) = (ran 𝐹 ∩ 𝐢)
3 eqid 2733 . . . . . . . . 9 (◑𝐹 β€œ 𝐢) = (◑𝐹 β€œ 𝐢)
4 eqid 2733 . . . . . . . . 9 (𝐹 β†Ύ (◑𝐹 β€œ 𝐢)) = (𝐹 β†Ύ (◑𝐹 β€œ 𝐢))
5 simp2 1138 . . . . . . . . 9 ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) β†’ 𝐺:𝐢⟢𝐷)
6 eqid 2733 . . . . . . . . 9 (𝐺 β†Ύ (ran 𝐹 ∩ 𝐢)) = (𝐺 β†Ύ (ran 𝐹 ∩ 𝐢))
7 simp3 1139 . . . . . . . . 9 ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) β†’ ran 𝐹 = 𝐢)
81, 2, 3, 4, 5, 6, 7f1cof1blem 45784 . . . . . . . 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 766 . . . . . . . 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 6784 . . . . . . . 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 462 . . . . . . . . . 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 624 . . . . . . . . 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 482 . . . . . . 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 482 . . . . . . . . 9 (((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) ∧ (𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐢)–1-1→𝐷) β†’ 𝐹:𝐴⟢𝐡)
185adantr 482 . . . . . . . . 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 486 . . . . . . . . 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 45779 . . . . . . . 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 463 . . . . . . 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 770 . . . . . . . . . 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 486 . . . . . . . . . . 11 (((◑𝐹 β€œ 𝐢) = 𝐴 ∧ (ran 𝐹 ∩ 𝐢) = 𝐢) β†’ (ran 𝐹 ∩ 𝐢) = 𝐢)
2423adantr 482 . . . . . . . . . 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 2734 . . . . . . . . . 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 6826 . . . . . . . . 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 772 . . . . . . . . . 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 766 . . . . . . . . . 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 6826 . . . . . . . . 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 610 . . . . . . 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 593 . . . . 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 6732 . . . . . . . . . . . 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 631 . . . . . . . . . 10 (𝐹:𝐴⟢𝐡 β†’ ((𝐹:𝐴⟢𝐡 ∧ Fun ◑𝐹) ↔ (𝐹:𝐴⟢ran 𝐹 ∧ Fun ◑𝐹)))
39 df-f1 6549 . . . . . . . . . 10 (𝐹:𝐴–1-1→𝐡 ↔ (𝐹:𝐴⟢𝐡 ∧ Fun ◑𝐹))
40 df-f1 6549 . . . . . . . . . 10 (𝐹:𝐴–1-1β†’ran 𝐹 ↔ (𝐹:𝐴⟢ran 𝐹 ∧ Fun ◑𝐹))
4138, 39, 403bitr4g 314 . . . . . . . . 9 (𝐹:𝐴⟢𝐡 β†’ (𝐹:𝐴–1-1→𝐡 ↔ 𝐹:𝐴–1-1β†’ran 𝐹))
42413ad2ant1 1134 . . . . . . . 8 ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) β†’ (𝐹:𝐴–1-1→𝐡 ↔ 𝐹:𝐴–1-1β†’ran 𝐹))
43 f1eq3 6785 . . . . . . . . 9 (ran 𝐹 = 𝐢 β†’ (𝐹:𝐴–1-1β†’ran 𝐹 ↔ 𝐹:𝐴–1-1→𝐢))
44433ad2ant3 1136 . . . . . . . 8 ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) β†’ (𝐹:𝐴–1-1β†’ran 𝐹 ↔ 𝐹:𝐴–1-1→𝐢))
4542, 44bitrd 279 . . . . . . 7 ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) β†’ (𝐹:𝐴–1-1→𝐡 ↔ 𝐹:𝐴–1-1→𝐢))
4645anbi2d 630 . . . . . 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 482 . . . . 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 257 . . . 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 463 . . 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 414 . 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 6799 . . . 4 ((𝐺:𝐢–1-1→𝐷 ∧ 𝐹:𝐴–1-1→𝐡) β†’ (𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐢)–1-1→𝐷)
5251ancoms 460 . . 3 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐺:𝐢–1-1→𝐷) β†’ (𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐢)–1-1→𝐷)
53 imaeq2 6056 . . . . . . . 8 (𝐢 = ran 𝐹 β†’ (◑𝐹 β€œ 𝐢) = (◑𝐹 β€œ ran 𝐹))
54 cnvimarndm 6082 . . . . . . . 8 (◑𝐹 β€œ ran 𝐹) = dom 𝐹
5553, 54eqtrdi 2789 . . . . . . 7 (𝐢 = ran 𝐹 β†’ (◑𝐹 β€œ 𝐢) = dom 𝐹)
5655eqcoms 2741 . . . . . 6 (ran 𝐹 = 𝐢 β†’ (◑𝐹 β€œ 𝐢) = dom 𝐹)
57563ad2ant3 1136 . . . . 5 ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) β†’ (◑𝐹 β€œ 𝐢) = dom 𝐹)
581fdmd 6729 . . . . 5 ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐢⟢𝐷 ∧ ran 𝐹 = 𝐢) β†’ dom 𝐹 = 𝐴)
5957, 58eqtrd 2773 . . . 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 397   ∧ w3a 1088   = wceq 1542   ∩ cin 3948  β—‘ccnv 5676  dom cdm 5677  ran crn 5678   β†Ύ cres 5679   β€œ cima 5680   ∘ ccom 5681  Fun wfun 6538  βŸΆwf 6540  β€“1-1β†’wf1 6541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  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-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-fv 6552
This theorem is referenced by:  f1ocof1ob  45789
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