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

Theorem konigthlem 9982
Description: Lemma for konigth 9983. (Contributed by Mario Carneiro, 22-Feb-2013.)
Hypotheses
Ref Expression
konigth.1 𝐴 ∈ V
konigth.2 𝑆 = 𝑖𝐴 (𝑀𝑖)
konigth.3 𝑃 = X𝑖𝐴 (𝑁𝑖)
konigth.4 𝐷 = (𝑖𝐴 ↦ (𝑎 ∈ (𝑀𝑖) ↦ ((𝑓𝑎)‘𝑖)))
konigth.5 𝐸 = (𝑖𝐴 ↦ (𝑒𝑖))
Assertion
Ref Expression
konigthlem (∀𝑖𝐴 (𝑀𝑖) ≺ (𝑁𝑖) → 𝑆𝑃)
Distinct variable groups:   𝐴,𝑎,𝑒,𝑓,𝑖   𝐷,𝑎,𝑒   𝐸,𝑎,𝑖   𝑀,𝑎,𝑓   𝑁,𝑎,𝑒,𝑓   𝑃,𝑎,𝑒,𝑓   𝑆,𝑎,𝑒,𝑓
Allowed substitution hints:   𝐷(𝑓,𝑖)   𝑃(𝑖)   𝑆(𝑖)   𝐸(𝑒,𝑓)   𝑀(𝑒,𝑖)   𝑁(𝑖)

Proof of Theorem konigthlem
StepHypRef Expression
1 fvex 6676 . . . . . . . . 9 (𝑀𝑖) ∈ V
2 fvex 6676 . . . . . . . . . . 11 ((𝑓𝑎)‘𝑖) ∈ V
3 eqid 2819 . . . . . . . . . . 11 (𝑎 ∈ (𝑀𝑖) ↦ ((𝑓𝑎)‘𝑖)) = (𝑎 ∈ (𝑀𝑖) ↦ ((𝑓𝑎)‘𝑖))
42, 3fnmpti 6484 . . . . . . . . . 10 (𝑎 ∈ (𝑀𝑖) ↦ ((𝑓𝑎)‘𝑖)) Fn (𝑀𝑖)
51mptex 6978 . . . . . . . . . . . 12 (𝑎 ∈ (𝑀𝑖) ↦ ((𝑓𝑎)‘𝑖)) ∈ V
6 konigth.4 . . . . . . . . . . . . 13 𝐷 = (𝑖𝐴 ↦ (𝑎 ∈ (𝑀𝑖) ↦ ((𝑓𝑎)‘𝑖)))
76fvmpt2 6772 . . . . . . . . . . . 12 ((𝑖𝐴 ∧ (𝑎 ∈ (𝑀𝑖) ↦ ((𝑓𝑎)‘𝑖)) ∈ V) → (𝐷𝑖) = (𝑎 ∈ (𝑀𝑖) ↦ ((𝑓𝑎)‘𝑖)))
85, 7mpan2 689 . . . . . . . . . . 11 (𝑖𝐴 → (𝐷𝑖) = (𝑎 ∈ (𝑀𝑖) ↦ ((𝑓𝑎)‘𝑖)))
98fneq1d 6439 . . . . . . . . . 10 (𝑖𝐴 → ((𝐷𝑖) Fn (𝑀𝑖) ↔ (𝑎 ∈ (𝑀𝑖) ↦ ((𝑓𝑎)‘𝑖)) Fn (𝑀𝑖)))
104, 9mpbiri 260 . . . . . . . . 9 (𝑖𝐴 → (𝐷𝑖) Fn (𝑀𝑖))
11 fnrndomg 9950 . . . . . . . . 9 ((𝑀𝑖) ∈ V → ((𝐷𝑖) Fn (𝑀𝑖) → ran (𝐷𝑖) ≼ (𝑀𝑖)))
121, 10, 11mpsyl 68 . . . . . . . 8 (𝑖𝐴 → ran (𝐷𝑖) ≼ (𝑀𝑖))
13 domsdomtr 8644 . . . . . . . 8 ((ran (𝐷𝑖) ≼ (𝑀𝑖) ∧ (𝑀𝑖) ≺ (𝑁𝑖)) → ran (𝐷𝑖) ≺ (𝑁𝑖))
1412, 13sylan 582 . . . . . . 7 ((𝑖𝐴 ∧ (𝑀𝑖) ≺ (𝑁𝑖)) → ran (𝐷𝑖) ≺ (𝑁𝑖))
15 sdomdif 8657 . . . . . . 7 (ran (𝐷𝑖) ≺ (𝑁𝑖) → ((𝑁𝑖) ∖ ran (𝐷𝑖)) ≠ ∅)
1614, 15syl 17 . . . . . 6 ((𝑖𝐴 ∧ (𝑀𝑖) ≺ (𝑁𝑖)) → ((𝑁𝑖) ∖ ran (𝐷𝑖)) ≠ ∅)
1716ralimiaa 3157 . . . . 5 (∀𝑖𝐴 (𝑀𝑖) ≺ (𝑁𝑖) → ∀𝑖𝐴 ((𝑁𝑖) ∖ ran (𝐷𝑖)) ≠ ∅)
18 konigth.1 . . . . . 6 𝐴 ∈ V
19 fvex 6676 . . . . . . 7 (𝑁𝑖) ∈ V
2019difexi 5223 . . . . . 6 ((𝑁𝑖) ∖ ran (𝐷𝑖)) ∈ V
2118, 20ac6c5 9896 . . . . 5 (∀𝑖𝐴 ((𝑁𝑖) ∖ ran (𝐷𝑖)) ≠ ∅ → ∃𝑒𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)))
22 equid 2012 . . . . . . 7 𝑓 = 𝑓
23 eldifi 4101 . . . . . . . . . . . . 13 ((𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → (𝑒𝑖) ∈ (𝑁𝑖))
24 fvex 6676 . . . . . . . . . . . . . . 15 (𝑒𝑖) ∈ V
25 konigth.5 . . . . . . . . . . . . . . . 16 𝐸 = (𝑖𝐴 ↦ (𝑒𝑖))
2625fvmpt2 6772 . . . . . . . . . . . . . . 15 ((𝑖𝐴 ∧ (𝑒𝑖) ∈ V) → (𝐸𝑖) = (𝑒𝑖))
2724, 26mpan2 689 . . . . . . . . . . . . . 14 (𝑖𝐴 → (𝐸𝑖) = (𝑒𝑖))
2827eleq1d 2895 . . . . . . . . . . . . 13 (𝑖𝐴 → ((𝐸𝑖) ∈ (𝑁𝑖) ↔ (𝑒𝑖) ∈ (𝑁𝑖)))
2923, 28syl5ibr 248 . . . . . . . . . . . 12 (𝑖𝐴 → ((𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → (𝐸𝑖) ∈ (𝑁𝑖)))
3029ralimia 3156 . . . . . . . . . . 11 (∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → ∀𝑖𝐴 (𝐸𝑖) ∈ (𝑁𝑖))
3124, 25fnmpti 6484 . . . . . . . . . . 11 𝐸 Fn 𝐴
3230, 31jctil 522 . . . . . . . . . 10 (∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → (𝐸 Fn 𝐴 ∧ ∀𝑖𝐴 (𝐸𝑖) ∈ (𝑁𝑖)))
3318mptex 6978 . . . . . . . . . . . 12 (𝑖𝐴 ↦ (𝑒𝑖)) ∈ V
3425, 33eqeltri 2907 . . . . . . . . . . 11 𝐸 ∈ V
3534elixp 8460 . . . . . . . . . 10 (𝐸X𝑖𝐴 (𝑁𝑖) ↔ (𝐸 Fn 𝐴 ∧ ∀𝑖𝐴 (𝐸𝑖) ∈ (𝑁𝑖)))
3632, 35sylibr 236 . . . . . . . . 9 (∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → 𝐸X𝑖𝐴 (𝑁𝑖))
37 konigth.3 . . . . . . . . 9 𝑃 = X𝑖𝐴 (𝑁𝑖)
3836, 37eleqtrrdi 2922 . . . . . . . 8 (∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → 𝐸𝑃)
39 foelrn 6865 . . . . . . . . . 10 ((𝑓:𝑆onto𝑃𝐸𝑃) → ∃𝑎𝑆 𝐸 = (𝑓𝑎))
4039expcom 416 . . . . . . . . 9 (𝐸𝑃 → (𝑓:𝑆onto𝑃 → ∃𝑎𝑆 𝐸 = (𝑓𝑎)))
41 konigth.2 . . . . . . . . . . . . . . 15 𝑆 = 𝑖𝐴 (𝑀𝑖)
4241eleq2i 2902 . . . . . . . . . . . . . 14 (𝑎𝑆𝑎 𝑖𝐴 (𝑀𝑖))
43 eliun 4914 . . . . . . . . . . . . . 14 (𝑎 𝑖𝐴 (𝑀𝑖) ↔ ∃𝑖𝐴 𝑎 ∈ (𝑀𝑖))
4442, 43bitri 277 . . . . . . . . . . . . 13 (𝑎𝑆 ↔ ∃𝑖𝐴 𝑎 ∈ (𝑀𝑖))
45 nfra1 3217 . . . . . . . . . . . . . . 15 𝑖𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖))
46 nfv 1908 . . . . . . . . . . . . . . 15 𝑖 𝐸 = (𝑓𝑎)
4745, 46nfan 1893 . . . . . . . . . . . . . 14 𝑖(∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) ∧ 𝐸 = (𝑓𝑎))
48 nfv 1908 . . . . . . . . . . . . . 14 𝑖 ¬ 𝑓 = 𝑓
4927ad2antrl 726 . . . . . . . . . . . . . . . . . . . 20 ((𝐸 = (𝑓𝑎) ∧ (𝑖𝐴𝑎 ∈ (𝑀𝑖))) → (𝐸𝑖) = (𝑒𝑖))
50 fveq1 6662 . . . . . . . . . . . . . . . . . . . . 21 (𝐸 = (𝑓𝑎) → (𝐸𝑖) = ((𝑓𝑎)‘𝑖))
518fveq1d 6665 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖𝐴 → ((𝐷𝑖)‘𝑎) = ((𝑎 ∈ (𝑀𝑖) ↦ ((𝑓𝑎)‘𝑖))‘𝑎))
523fvmpt2 6772 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑎 ∈ (𝑀𝑖) ∧ ((𝑓𝑎)‘𝑖) ∈ V) → ((𝑎 ∈ (𝑀𝑖) ↦ ((𝑓𝑎)‘𝑖))‘𝑎) = ((𝑓𝑎)‘𝑖))
532, 52mpan2 689 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 ∈ (𝑀𝑖) → ((𝑎 ∈ (𝑀𝑖) ↦ ((𝑓𝑎)‘𝑖))‘𝑎) = ((𝑓𝑎)‘𝑖))
5451, 53sylan9eq 2874 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖𝐴𝑎 ∈ (𝑀𝑖)) → ((𝐷𝑖)‘𝑎) = ((𝑓𝑎)‘𝑖))
5554eqcomd 2825 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖𝐴𝑎 ∈ (𝑀𝑖)) → ((𝑓𝑎)‘𝑖) = ((𝐷𝑖)‘𝑎))
5650, 55sylan9eq 2874 . . . . . . . . . . . . . . . . . . . 20 ((𝐸 = (𝑓𝑎) ∧ (𝑖𝐴𝑎 ∈ (𝑀𝑖))) → (𝐸𝑖) = ((𝐷𝑖)‘𝑎))
5749, 56eqtr3d 2856 . . . . . . . . . . . . . . . . . . 19 ((𝐸 = (𝑓𝑎) ∧ (𝑖𝐴𝑎 ∈ (𝑀𝑖))) → (𝑒𝑖) = ((𝐷𝑖)‘𝑎))
58 fnfvelrn 6841 . . . . . . . . . . . . . . . . . . . . 21 (((𝐷𝑖) Fn (𝑀𝑖) ∧ 𝑎 ∈ (𝑀𝑖)) → ((𝐷𝑖)‘𝑎) ∈ ran (𝐷𝑖))
5910, 58sylan 582 . . . . . . . . . . . . . . . . . . . 20 ((𝑖𝐴𝑎 ∈ (𝑀𝑖)) → ((𝐷𝑖)‘𝑎) ∈ ran (𝐷𝑖))
6059adantl 484 . . . . . . . . . . . . . . . . . . 19 ((𝐸 = (𝑓𝑎) ∧ (𝑖𝐴𝑎 ∈ (𝑀𝑖))) → ((𝐷𝑖)‘𝑎) ∈ ran (𝐷𝑖))
6157, 60eqeltrd 2911 . . . . . . . . . . . . . . . . . 18 ((𝐸 = (𝑓𝑎) ∧ (𝑖𝐴𝑎 ∈ (𝑀𝑖))) → (𝑒𝑖) ∈ ran (𝐷𝑖))
62613adant1 1124 . . . . . . . . . . . . . . . . 17 ((∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) ∧ 𝐸 = (𝑓𝑎) ∧ (𝑖𝐴𝑎 ∈ (𝑀𝑖))) → (𝑒𝑖) ∈ ran (𝐷𝑖))
63 simp1 1130 . . . . . . . . . . . . . . . . . 18 ((∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) ∧ 𝐸 = (𝑓𝑎) ∧ (𝑖𝐴𝑎 ∈ (𝑀𝑖))) → ∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)))
64 simp3l 1195 . . . . . . . . . . . . . . . . . 18 ((∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) ∧ 𝐸 = (𝑓𝑎) ∧ (𝑖𝐴𝑎 ∈ (𝑀𝑖))) → 𝑖𝐴)
65 rsp 3203 . . . . . . . . . . . . . . . . . . 19 (∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → (𝑖𝐴 → (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖))))
66 eldifn 4102 . . . . . . . . . . . . . . . . . . 19 ((𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → ¬ (𝑒𝑖) ∈ ran (𝐷𝑖))
6765, 66syl6 35 . . . . . . . . . . . . . . . . . 18 (∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → (𝑖𝐴 → ¬ (𝑒𝑖) ∈ ran (𝐷𝑖)))
6863, 64, 67sylc 65 . . . . . . . . . . . . . . . . 17 ((∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) ∧ 𝐸 = (𝑓𝑎) ∧ (𝑖𝐴𝑎 ∈ (𝑀𝑖))) → ¬ (𝑒𝑖) ∈ ran (𝐷𝑖))
6962, 68pm2.21dd 197 . . . . . . . . . . . . . . . 16 ((∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) ∧ 𝐸 = (𝑓𝑎) ∧ (𝑖𝐴𝑎 ∈ (𝑀𝑖))) → ¬ 𝑓 = 𝑓)
70693expia 1115 . . . . . . . . . . . . . . 15 ((∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) ∧ 𝐸 = (𝑓𝑎)) → ((𝑖𝐴𝑎 ∈ (𝑀𝑖)) → ¬ 𝑓 = 𝑓))
7170expd 418 . . . . . . . . . . . . . 14 ((∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) ∧ 𝐸 = (𝑓𝑎)) → (𝑖𝐴 → (𝑎 ∈ (𝑀𝑖) → ¬ 𝑓 = 𝑓)))
7247, 48, 71rexlimd 3315 . . . . . . . . . . . . 13 ((∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) ∧ 𝐸 = (𝑓𝑎)) → (∃𝑖𝐴 𝑎 ∈ (𝑀𝑖) → ¬ 𝑓 = 𝑓))
7344, 72syl5bi 244 . . . . . . . . . . . 12 ((∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) ∧ 𝐸 = (𝑓𝑎)) → (𝑎𝑆 → ¬ 𝑓 = 𝑓))
7473ex 415 . . . . . . . . . . 11 (∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → (𝐸 = (𝑓𝑎) → (𝑎𝑆 → ¬ 𝑓 = 𝑓)))
7574com23 86 . . . . . . . . . 10 (∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → (𝑎𝑆 → (𝐸 = (𝑓𝑎) → ¬ 𝑓 = 𝑓)))
7675rexlimdv 3281 . . . . . . . . 9 (∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → (∃𝑎𝑆 𝐸 = (𝑓𝑎) → ¬ 𝑓 = 𝑓))
7740, 76syl9r 78 . . . . . . . 8 (∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → (𝐸𝑃 → (𝑓:𝑆onto𝑃 → ¬ 𝑓 = 𝑓)))
7838, 77mpd 15 . . . . . . 7 (∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → (𝑓:𝑆onto𝑃 → ¬ 𝑓 = 𝑓))
7922, 78mt2i 139 . . . . . 6 (∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → ¬ 𝑓:𝑆onto𝑃)
8079exlimiv 1924 . . . . 5 (∃𝑒𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → ¬ 𝑓:𝑆onto𝑃)
8117, 21, 803syl 18 . . . 4 (∀𝑖𝐴 (𝑀𝑖) ≺ (𝑁𝑖) → ¬ 𝑓:𝑆onto𝑃)
8281nexdv 1930 . . 3 (∀𝑖𝐴 (𝑀𝑖) ≺ (𝑁𝑖) → ¬ ∃𝑓 𝑓:𝑆onto𝑃)
8310dom 8639 . . . . . . . 8 ∅ ≼ (𝑀𝑖)
84 domsdomtr 8644 . . . . . . . 8 ((∅ ≼ (𝑀𝑖) ∧ (𝑀𝑖) ≺ (𝑁𝑖)) → ∅ ≺ (𝑁𝑖))
8583, 84mpan 688 . . . . . . 7 ((𝑀𝑖) ≺ (𝑁𝑖) → ∅ ≺ (𝑁𝑖))
86190sdom 8640 . . . . . . 7 (∅ ≺ (𝑁𝑖) ↔ (𝑁𝑖) ≠ ∅)
8785, 86sylib 220 . . . . . 6 ((𝑀𝑖) ≺ (𝑁𝑖) → (𝑁𝑖) ≠ ∅)
8887ralimi 3158 . . . . 5 (∀𝑖𝐴 (𝑀𝑖) ≺ (𝑁𝑖) → ∀𝑖𝐴 (𝑁𝑖) ≠ ∅)
8937neeq1i 3078 . . . . . 6 (𝑃 ≠ ∅ ↔ X𝑖𝐴 (𝑁𝑖) ≠ ∅)
9019rgenw 3148 . . . . . . . . 9 𝑖𝐴 (𝑁𝑖) ∈ V
91 ixpexg 8478 . . . . . . . . 9 (∀𝑖𝐴 (𝑁𝑖) ∈ V → X𝑖𝐴 (𝑁𝑖) ∈ V)
9290, 91ax-mp 5 . . . . . . . 8 X𝑖𝐴 (𝑁𝑖) ∈ V
9337, 92eqeltri 2907 . . . . . . 7 𝑃 ∈ V
94930sdom 8640 . . . . . 6 (∅ ≺ 𝑃𝑃 ≠ ∅)
9518, 19ac9 9897 . . . . . 6 (∀𝑖𝐴 (𝑁𝑖) ≠ ∅ ↔ X𝑖𝐴 (𝑁𝑖) ≠ ∅)
9689, 94, 953bitr4i 305 . . . . 5 (∅ ≺ 𝑃 ↔ ∀𝑖𝐴 (𝑁𝑖) ≠ ∅)
9788, 96sylibr 236 . . . 4 (∀𝑖𝐴 (𝑀𝑖) ≺ (𝑁𝑖) → ∅ ≺ 𝑃)
9818, 1iunex 7661 . . . . . . 7 𝑖𝐴 (𝑀𝑖) ∈ V
9941, 98eqeltri 2907 . . . . . 6 𝑆 ∈ V
100 domtri 9970 . . . . . 6 ((𝑃 ∈ V ∧ 𝑆 ∈ V) → (𝑃𝑆 ↔ ¬ 𝑆𝑃))
10193, 99, 100mp2an 690 . . . . 5 (𝑃𝑆 ↔ ¬ 𝑆𝑃)
102101biimpri 230 . . . 4 𝑆𝑃𝑃𝑆)
103 fodomr 8660 . . . 4 ((∅ ≺ 𝑃𝑃𝑆) → ∃𝑓 𝑓:𝑆onto𝑃)
10497, 102, 103syl2an 597 . . 3 ((∀𝑖𝐴 (𝑀𝑖) ≺ (𝑁𝑖) ∧ ¬ 𝑆𝑃) → ∃𝑓 𝑓:𝑆onto𝑃)
10582, 104mtand 814 . 2 (∀𝑖𝐴 (𝑀𝑖) ≺ (𝑁𝑖) → ¬ ¬ 𝑆𝑃)
106105notnotrd 135 1 (∀𝑖𝐴 (𝑀𝑖) ≺ (𝑁𝑖) → 𝑆𝑃)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  w3a 1081   = wceq 1530  wex 1773  wcel 2107  wne 3014  wral 3136  wrex 3137  Vcvv 3493  cdif 3931  c0 4289   ciun 4910   class class class wbr 5057  cmpt 5137  ran crn 5549   Fn wfn 6343  ontowfo 6346  cfv 6348  Xcixp 8453  cdom 8499  csdm 8500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-ac2 9877
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-er 8281  df-map 8400  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-card 9360  df-acn 9363  df-ac 9534
This theorem is referenced by:  konigth  9983
  Copyright terms: Public domain W3C validator