MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfac9 Structured version   Visualization version   GIF version

Theorem dfac9 10179
Description: Equivalence of the axiom of choice with a statement related to ac9 10526; definition AC3 of [Schechter] p. 139. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
dfac9 (CHOICE ↔ ∀𝑓((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓𝑥) ≠ ∅))
Distinct variable group:   𝑥,𝑓

Proof of Theorem dfac9
Dummy variables 𝑔 𝑠 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfac3 10164 . 2 (CHOICE ↔ ∀𝑠𝑔𝑡𝑠 (𝑡 ≠ ∅ → (𝑔𝑡) ∈ 𝑡))
2 vex 3466 . . . . . . 7 𝑓 ∈ V
32rnex 7923 . . . . . 6 ran 𝑓 ∈ V
4 raleq 3312 . . . . . . 7 (𝑠 = ran 𝑓 → (∀𝑡𝑠 (𝑡 ≠ ∅ → (𝑔𝑡) ∈ 𝑡) ↔ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔𝑡) ∈ 𝑡)))
54exbidv 1917 . . . . . 6 (𝑠 = ran 𝑓 → (∃𝑔𝑡𝑠 (𝑡 ≠ ∅ → (𝑔𝑡) ∈ 𝑡) ↔ ∃𝑔𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔𝑡) ∈ 𝑡)))
63, 5spcv 3591 . . . . 5 (∀𝑠𝑔𝑡𝑠 (𝑡 ≠ ∅ → (𝑔𝑡) ∈ 𝑡) → ∃𝑔𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔𝑡) ∈ 𝑡))
7 df-nel 3037 . . . . . . . . . . . . . . 15 (∅ ∉ ran 𝑓 ↔ ¬ ∅ ∈ ran 𝑓)
87biimpi 215 . . . . . . . . . . . . . 14 (∅ ∉ ran 𝑓 → ¬ ∅ ∈ ran 𝑓)
98ad2antlr 725 . . . . . . . . . . . . 13 (((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ 𝑥 ∈ dom 𝑓) → ¬ ∅ ∈ ran 𝑓)
10 fvelrn 7090 . . . . . . . . . . . . . . . 16 ((Fun 𝑓𝑥 ∈ dom 𝑓) → (𝑓𝑥) ∈ ran 𝑓)
1110adantlr 713 . . . . . . . . . . . . . . 15 (((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ 𝑥 ∈ dom 𝑓) → (𝑓𝑥) ∈ ran 𝑓)
12 eleq1 2814 . . . . . . . . . . . . . . 15 ((𝑓𝑥) = ∅ → ((𝑓𝑥) ∈ ran 𝑓 ↔ ∅ ∈ ran 𝑓))
1311, 12syl5ibcom 244 . . . . . . . . . . . . . 14 (((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ 𝑥 ∈ dom 𝑓) → ((𝑓𝑥) = ∅ → ∅ ∈ ran 𝑓))
1413necon3bd 2944 . . . . . . . . . . . . 13 (((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ 𝑥 ∈ dom 𝑓) → (¬ ∅ ∈ ran 𝑓 → (𝑓𝑥) ≠ ∅))
159, 14mpd 15 . . . . . . . . . . . 12 (((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ 𝑥 ∈ dom 𝑓) → (𝑓𝑥) ≠ ∅)
1615adantlr 713 . . . . . . . . . . 11 ((((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔𝑡) ∈ 𝑡)) ∧ 𝑥 ∈ dom 𝑓) → (𝑓𝑥) ≠ ∅)
17 neeq1 2993 . . . . . . . . . . . . 13 (𝑡 = (𝑓𝑥) → (𝑡 ≠ ∅ ↔ (𝑓𝑥) ≠ ∅))
18 fveq2 6901 . . . . . . . . . . . . . 14 (𝑡 = (𝑓𝑥) → (𝑔𝑡) = (𝑔‘(𝑓𝑥)))
19 id 22 . . . . . . . . . . . . . 14 (𝑡 = (𝑓𝑥) → 𝑡 = (𝑓𝑥))
2018, 19eleq12d 2820 . . . . . . . . . . . . 13 (𝑡 = (𝑓𝑥) → ((𝑔𝑡) ∈ 𝑡 ↔ (𝑔‘(𝑓𝑥)) ∈ (𝑓𝑥)))
2117, 20imbi12d 343 . . . . . . . . . . . 12 (𝑡 = (𝑓𝑥) → ((𝑡 ≠ ∅ → (𝑔𝑡) ∈ 𝑡) ↔ ((𝑓𝑥) ≠ ∅ → (𝑔‘(𝑓𝑥)) ∈ (𝑓𝑥))))
22 simplr 767 . . . . . . . . . . . 12 ((((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔𝑡) ∈ 𝑡)) ∧ 𝑥 ∈ dom 𝑓) → ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔𝑡) ∈ 𝑡))
2310ad4ant14 750 . . . . . . . . . . . 12 ((((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔𝑡) ∈ 𝑡)) ∧ 𝑥 ∈ dom 𝑓) → (𝑓𝑥) ∈ ran 𝑓)
2421, 22, 23rspcdva 3609 . . . . . . . . . . 11 ((((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔𝑡) ∈ 𝑡)) ∧ 𝑥 ∈ dom 𝑓) → ((𝑓𝑥) ≠ ∅ → (𝑔‘(𝑓𝑥)) ∈ (𝑓𝑥)))
2516, 24mpd 15 . . . . . . . . . 10 ((((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔𝑡) ∈ 𝑡)) ∧ 𝑥 ∈ dom 𝑓) → (𝑔‘(𝑓𝑥)) ∈ (𝑓𝑥))
2625ralrimiva 3136 . . . . . . . . 9 (((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔𝑡) ∈ 𝑡)) → ∀𝑥 ∈ dom 𝑓(𝑔‘(𝑓𝑥)) ∈ (𝑓𝑥))
272dmex 7922 . . . . . . . . . 10 dom 𝑓 ∈ V
28 mptelixpg 8964 . . . . . . . . . 10 (dom 𝑓 ∈ V → ((𝑥 ∈ dom 𝑓 ↦ (𝑔‘(𝑓𝑥))) ∈ X𝑥 ∈ dom 𝑓(𝑓𝑥) ↔ ∀𝑥 ∈ dom 𝑓(𝑔‘(𝑓𝑥)) ∈ (𝑓𝑥)))
2927, 28ax-mp 5 . . . . . . . . 9 ((𝑥 ∈ dom 𝑓 ↦ (𝑔‘(𝑓𝑥))) ∈ X𝑥 ∈ dom 𝑓(𝑓𝑥) ↔ ∀𝑥 ∈ dom 𝑓(𝑔‘(𝑓𝑥)) ∈ (𝑓𝑥))
3026, 29sylibr 233 . . . . . . . 8 (((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔𝑡) ∈ 𝑡)) → (𝑥 ∈ dom 𝑓 ↦ (𝑔‘(𝑓𝑥))) ∈ X𝑥 ∈ dom 𝑓(𝑓𝑥))
3130ne0d 4338 . . . . . . 7 (((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔𝑡) ∈ 𝑡)) → X𝑥 ∈ dom 𝑓(𝑓𝑥) ≠ ∅)
3231ex 411 . . . . . 6 ((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → (∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔𝑡) ∈ 𝑡) → X𝑥 ∈ dom 𝑓(𝑓𝑥) ≠ ∅))
3332exlimdv 1929 . . . . 5 ((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → (∃𝑔𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔𝑡) ∈ 𝑡) → X𝑥 ∈ dom 𝑓(𝑓𝑥) ≠ ∅))
346, 33syl5com 31 . . . 4 (∀𝑠𝑔𝑡𝑠 (𝑡 ≠ ∅ → (𝑔𝑡) ∈ 𝑡) → ((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓𝑥) ≠ ∅))
3534alrimiv 1923 . . 3 (∀𝑠𝑔𝑡𝑠 (𝑡 ≠ ∅ → (𝑔𝑡) ∈ 𝑡) → ∀𝑓((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓𝑥) ≠ ∅))
36 fnresi 6690 . . . . . . 7 ( I ↾ (𝑠 ∖ {∅})) Fn (𝑠 ∖ {∅})
37 fnfun 6660 . . . . . . 7 (( I ↾ (𝑠 ∖ {∅})) Fn (𝑠 ∖ {∅}) → Fun ( I ↾ (𝑠 ∖ {∅})))
3836, 37ax-mp 5 . . . . . 6 Fun ( I ↾ (𝑠 ∖ {∅}))
39 neldifsn 4801 . . . . . 6 ¬ ∅ ∈ (𝑠 ∖ {∅})
40 vex 3466 . . . . . . . . 9 𝑠 ∈ V
4140difexi 5335 . . . . . . . 8 (𝑠 ∖ {∅}) ∈ V
42 resiexg 7925 . . . . . . . 8 ((𝑠 ∖ {∅}) ∈ V → ( I ↾ (𝑠 ∖ {∅})) ∈ V)
4341, 42ax-mp 5 . . . . . . 7 ( I ↾ (𝑠 ∖ {∅})) ∈ V
44 funeq 6579 . . . . . . . . 9 (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → (Fun 𝑓 ↔ Fun ( I ↾ (𝑠 ∖ {∅}))))
45 rneq 5942 . . . . . . . . . . . . 13 (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → ran 𝑓 = ran ( I ↾ (𝑠 ∖ {∅})))
46 rnresi 6084 . . . . . . . . . . . . 13 ran ( I ↾ (𝑠 ∖ {∅})) = (𝑠 ∖ {∅})
4745, 46eqtrdi 2782 . . . . . . . . . . . 12 (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → ran 𝑓 = (𝑠 ∖ {∅}))
4847eleq2d 2812 . . . . . . . . . . 11 (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → (∅ ∈ ran 𝑓 ↔ ∅ ∈ (𝑠 ∖ {∅})))
4948notbid 317 . . . . . . . . . 10 (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → (¬ ∅ ∈ ran 𝑓 ↔ ¬ ∅ ∈ (𝑠 ∖ {∅})))
507, 49bitrid 282 . . . . . . . . 9 (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → (∅ ∉ ran 𝑓 ↔ ¬ ∅ ∈ (𝑠 ∖ {∅})))
5144, 50anbi12d 630 . . . . . . . 8 (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → ((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ↔ (Fun ( I ↾ (𝑠 ∖ {∅})) ∧ ¬ ∅ ∈ (𝑠 ∖ {∅}))))
52 dmeq 5910 . . . . . . . . . . . 12 (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → dom 𝑓 = dom ( I ↾ (𝑠 ∖ {∅})))
53 dmresi 6061 . . . . . . . . . . . 12 dom ( I ↾ (𝑠 ∖ {∅})) = (𝑠 ∖ {∅})
5452, 53eqtrdi 2782 . . . . . . . . . . 11 (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → dom 𝑓 = (𝑠 ∖ {∅}))
5554ixpeq1d 8938 . . . . . . . . . 10 (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → X𝑥 ∈ dom 𝑓(𝑓𝑥) = X𝑥 ∈ (𝑠 ∖ {∅})(𝑓𝑥))
56 fveq1 6900 . . . . . . . . . . . 12 (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → (𝑓𝑥) = (( I ↾ (𝑠 ∖ {∅}))‘𝑥))
57 fvresi 7187 . . . . . . . . . . . 12 (𝑥 ∈ (𝑠 ∖ {∅}) → (( I ↾ (𝑠 ∖ {∅}))‘𝑥) = 𝑥)
5856, 57sylan9eq 2786 . . . . . . . . . . 11 ((𝑓 = ( I ↾ (𝑠 ∖ {∅})) ∧ 𝑥 ∈ (𝑠 ∖ {∅})) → (𝑓𝑥) = 𝑥)
5958ixpeq2dva 8941 . . . . . . . . . 10 (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → X𝑥 ∈ (𝑠 ∖ {∅})(𝑓𝑥) = X𝑥 ∈ (𝑠 ∖ {∅})𝑥)
6055, 59eqtrd 2766 . . . . . . . . 9 (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → X𝑥 ∈ dom 𝑓(𝑓𝑥) = X𝑥 ∈ (𝑠 ∖ {∅})𝑥)
6160neeq1d 2990 . . . . . . . 8 (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → (X𝑥 ∈ dom 𝑓(𝑓𝑥) ≠ ∅ ↔ X𝑥 ∈ (𝑠 ∖ {∅})𝑥 ≠ ∅))
6251, 61imbi12d 343 . . . . . . 7 (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → (((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓𝑥) ≠ ∅) ↔ ((Fun ( I ↾ (𝑠 ∖ {∅})) ∧ ¬ ∅ ∈ (𝑠 ∖ {∅})) → X𝑥 ∈ (𝑠 ∖ {∅})𝑥 ≠ ∅)))
6343, 62spcv 3591 . . . . . 6 (∀𝑓((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓𝑥) ≠ ∅) → ((Fun ( I ↾ (𝑠 ∖ {∅})) ∧ ¬ ∅ ∈ (𝑠 ∖ {∅})) → X𝑥 ∈ (𝑠 ∖ {∅})𝑥 ≠ ∅))
6438, 39, 63mp2ani 696 . . . . 5 (∀𝑓((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓𝑥) ≠ ∅) → X𝑥 ∈ (𝑠 ∖ {∅})𝑥 ≠ ∅)
65 n0 4349 . . . . . 6 (X𝑥 ∈ (𝑠 ∖ {∅})𝑥 ≠ ∅ ↔ ∃𝑔 𝑔X𝑥 ∈ (𝑠 ∖ {∅})𝑥)
66 vex 3466 . . . . . . . . 9 𝑔 ∈ V
6766elixp 8933 . . . . . . . 8 (𝑔X𝑥 ∈ (𝑠 ∖ {∅})𝑥 ↔ (𝑔 Fn (𝑠 ∖ {∅}) ∧ ∀𝑥 ∈ (𝑠 ∖ {∅})(𝑔𝑥) ∈ 𝑥))
68 eldifsn 4795 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑠 ∖ {∅}) ↔ (𝑥𝑠𝑥 ≠ ∅))
6968imbi1i 348 . . . . . . . . . . . 12 ((𝑥 ∈ (𝑠 ∖ {∅}) → (𝑔𝑥) ∈ 𝑥) ↔ ((𝑥𝑠𝑥 ≠ ∅) → (𝑔𝑥) ∈ 𝑥))
70 impexp 449 . . . . . . . . . . . 12 (((𝑥𝑠𝑥 ≠ ∅) → (𝑔𝑥) ∈ 𝑥) ↔ (𝑥𝑠 → (𝑥 ≠ ∅ → (𝑔𝑥) ∈ 𝑥)))
7169, 70bitri 274 . . . . . . . . . . 11 ((𝑥 ∈ (𝑠 ∖ {∅}) → (𝑔𝑥) ∈ 𝑥) ↔ (𝑥𝑠 → (𝑥 ≠ ∅ → (𝑔𝑥) ∈ 𝑥)))
7271ralbii2 3079 . . . . . . . . . 10 (∀𝑥 ∈ (𝑠 ∖ {∅})(𝑔𝑥) ∈ 𝑥 ↔ ∀𝑥𝑠 (𝑥 ≠ ∅ → (𝑔𝑥) ∈ 𝑥))
73 neeq1 2993 . . . . . . . . . . . 12 (𝑥 = 𝑡 → (𝑥 ≠ ∅ ↔ 𝑡 ≠ ∅))
74 fveq2 6901 . . . . . . . . . . . . 13 (𝑥 = 𝑡 → (𝑔𝑥) = (𝑔𝑡))
75 id 22 . . . . . . . . . . . . 13 (𝑥 = 𝑡𝑥 = 𝑡)
7674, 75eleq12d 2820 . . . . . . . . . . . 12 (𝑥 = 𝑡 → ((𝑔𝑥) ∈ 𝑥 ↔ (𝑔𝑡) ∈ 𝑡))
7773, 76imbi12d 343 . . . . . . . . . . 11 (𝑥 = 𝑡 → ((𝑥 ≠ ∅ → (𝑔𝑥) ∈ 𝑥) ↔ (𝑡 ≠ ∅ → (𝑔𝑡) ∈ 𝑡)))
7877cbvralvw 3225 . . . . . . . . . 10 (∀𝑥𝑠 (𝑥 ≠ ∅ → (𝑔𝑥) ∈ 𝑥) ↔ ∀𝑡𝑠 (𝑡 ≠ ∅ → (𝑔𝑡) ∈ 𝑡))
7972, 78bitri 274 . . . . . . . . 9 (∀𝑥 ∈ (𝑠 ∖ {∅})(𝑔𝑥) ∈ 𝑥 ↔ ∀𝑡𝑠 (𝑡 ≠ ∅ → (𝑔𝑡) ∈ 𝑡))
8079biimpi 215 . . . . . . . 8 (∀𝑥 ∈ (𝑠 ∖ {∅})(𝑔𝑥) ∈ 𝑥 → ∀𝑡𝑠 (𝑡 ≠ ∅ → (𝑔𝑡) ∈ 𝑡))
8167, 80simplbiim 503 . . . . . . 7 (𝑔X𝑥 ∈ (𝑠 ∖ {∅})𝑥 → ∀𝑡𝑠 (𝑡 ≠ ∅ → (𝑔𝑡) ∈ 𝑡))
8281eximi 1830 . . . . . 6 (∃𝑔 𝑔X𝑥 ∈ (𝑠 ∖ {∅})𝑥 → ∃𝑔𝑡𝑠 (𝑡 ≠ ∅ → (𝑔𝑡) ∈ 𝑡))
8365, 82sylbi 216 . . . . 5 (X𝑥 ∈ (𝑠 ∖ {∅})𝑥 ≠ ∅ → ∃𝑔𝑡𝑠 (𝑡 ≠ ∅ → (𝑔𝑡) ∈ 𝑡))
8464, 83syl 17 . . . 4 (∀𝑓((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓𝑥) ≠ ∅) → ∃𝑔𝑡𝑠 (𝑡 ≠ ∅ → (𝑔𝑡) ∈ 𝑡))
8584alrimiv 1923 . . 3 (∀𝑓((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓𝑥) ≠ ∅) → ∀𝑠𝑔𝑡𝑠 (𝑡 ≠ ∅ → (𝑔𝑡) ∈ 𝑡))
8635, 85impbii 208 . 2 (∀𝑠𝑔𝑡𝑠 (𝑡 ≠ ∅ → (𝑔𝑡) ∈ 𝑡) ↔ ∀𝑓((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓𝑥) ≠ ∅))
871, 86bitri 274 1 (CHOICE ↔ ∀𝑓((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓𝑥) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wal 1532   = wceq 1534  wex 1774  wcel 2099  wne 2930  wnel 3036  wral 3051  Vcvv 3462  cdif 3944  c0 4325  {csn 4633  cmpt 5236   I cid 5579  dom cdm 5682  ran crn 5683  cres 5684  Fun wfun 6548   Fn wfn 6549  cfv 6554  Xcixp 8926  CHOICEwac 10158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ixp 8927  df-ac 10159
This theorem is referenced by:  dfac14  23613  dfac21  42727
  Copyright terms: Public domain W3C validator