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Theorem konigthlem 9428
 Description: Lemma for konigth 9429. (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 6239 . . . . . . . . 9 (𝑀𝑖) ∈ V
2 fvex 6239 . . . . . . . . . . 11 ((𝑓𝑎)‘𝑖) ∈ V
3 eqid 2651 . . . . . . . . . . 11 (𝑎 ∈ (𝑀𝑖) ↦ ((𝑓𝑎)‘𝑖)) = (𝑎 ∈ (𝑀𝑖) ↦ ((𝑓𝑎)‘𝑖))
42, 3fnmpti 6060 . . . . . . . . . 10 (𝑎 ∈ (𝑀𝑖) ↦ ((𝑓𝑎)‘𝑖)) Fn (𝑀𝑖)
51mptex 6527 . . . . . . . . . . . 12 (𝑎 ∈ (𝑀𝑖) ↦ ((𝑓𝑎)‘𝑖)) ∈ V
6 konigth.4 . . . . . . . . . . . . 13 𝐷 = (𝑖𝐴 ↦ (𝑎 ∈ (𝑀𝑖) ↦ ((𝑓𝑎)‘𝑖)))
76fvmpt2 6330 . . . . . . . . . . . 12 ((𝑖𝐴 ∧ (𝑎 ∈ (𝑀𝑖) ↦ ((𝑓𝑎)‘𝑖)) ∈ V) → (𝐷𝑖) = (𝑎 ∈ (𝑀𝑖) ↦ ((𝑓𝑎)‘𝑖)))
85, 7mpan2 707 . . . . . . . . . . 11 (𝑖𝐴 → (𝐷𝑖) = (𝑎 ∈ (𝑀𝑖) ↦ ((𝑓𝑎)‘𝑖)))
98fneq1d 6019 . . . . . . . . . 10 (𝑖𝐴 → ((𝐷𝑖) Fn (𝑀𝑖) ↔ (𝑎 ∈ (𝑀𝑖) ↦ ((𝑓𝑎)‘𝑖)) Fn (𝑀𝑖)))
104, 9mpbiri 248 . . . . . . . . 9 (𝑖𝐴 → (𝐷𝑖) Fn (𝑀𝑖))
11 fnrndomg 9396 . . . . . . . . 9 ((𝑀𝑖) ∈ V → ((𝐷𝑖) Fn (𝑀𝑖) → ran (𝐷𝑖) ≼ (𝑀𝑖)))
121, 10, 11mpsyl 68 . . . . . . . 8 (𝑖𝐴 → ran (𝐷𝑖) ≼ (𝑀𝑖))
13 domsdomtr 8136 . . . . . . . 8 ((ran (𝐷𝑖) ≼ (𝑀𝑖) ∧ (𝑀𝑖) ≺ (𝑁𝑖)) → ran (𝐷𝑖) ≺ (𝑁𝑖))
1412, 13sylan 487 . . . . . . 7 ((𝑖𝐴 ∧ (𝑀𝑖) ≺ (𝑁𝑖)) → ran (𝐷𝑖) ≺ (𝑁𝑖))
15 sdomdif 8149 . . . . . . 7 (ran (𝐷𝑖) ≺ (𝑁𝑖) → ((𝑁𝑖) ∖ ran (𝐷𝑖)) ≠ ∅)
1614, 15syl 17 . . . . . 6 ((𝑖𝐴 ∧ (𝑀𝑖) ≺ (𝑁𝑖)) → ((𝑁𝑖) ∖ ran (𝐷𝑖)) ≠ ∅)
1716ralimiaa 2980 . . . . 5 (∀𝑖𝐴 (𝑀𝑖) ≺ (𝑁𝑖) → ∀𝑖𝐴 ((𝑁𝑖) ∖ ran (𝐷𝑖)) ≠ ∅)
18 konigth.1 . . . . . 6 𝐴 ∈ V
19 fvex 6239 . . . . . . 7 (𝑁𝑖) ∈ V
20 difss 3770 . . . . . . 7 ((𝑁𝑖) ∖ ran (𝐷𝑖)) ⊆ (𝑁𝑖)
2119, 20ssexi 4836 . . . . . 6 ((𝑁𝑖) ∖ ran (𝐷𝑖)) ∈ V
2218, 21ac6c5 9342 . . . . 5 (∀𝑖𝐴 ((𝑁𝑖) ∖ ran (𝐷𝑖)) ≠ ∅ → ∃𝑒𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)))
23 equid 1985 . . . . . . 7 𝑓 = 𝑓
24 eldifi 3765 . . . . . . . . . . . . 13 ((𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → (𝑒𝑖) ∈ (𝑁𝑖))
25 fvex 6239 . . . . . . . . . . . . . . 15 (𝑒𝑖) ∈ V
26 konigth.5 . . . . . . . . . . . . . . . 16 𝐸 = (𝑖𝐴 ↦ (𝑒𝑖))
2726fvmpt2 6330 . . . . . . . . . . . . . . 15 ((𝑖𝐴 ∧ (𝑒𝑖) ∈ V) → (𝐸𝑖) = (𝑒𝑖))
2825, 27mpan2 707 . . . . . . . . . . . . . 14 (𝑖𝐴 → (𝐸𝑖) = (𝑒𝑖))
2928eleq1d 2715 . . . . . . . . . . . . 13 (𝑖𝐴 → ((𝐸𝑖) ∈ (𝑁𝑖) ↔ (𝑒𝑖) ∈ (𝑁𝑖)))
3024, 29syl5ibr 236 . . . . . . . . . . . 12 (𝑖𝐴 → ((𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → (𝐸𝑖) ∈ (𝑁𝑖)))
3130ralimia 2979 . . . . . . . . . . 11 (∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → ∀𝑖𝐴 (𝐸𝑖) ∈ (𝑁𝑖))
3225, 26fnmpti 6060 . . . . . . . . . . 11 𝐸 Fn 𝐴
3331, 32jctil 559 . . . . . . . . . 10 (∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → (𝐸 Fn 𝐴 ∧ ∀𝑖𝐴 (𝐸𝑖) ∈ (𝑁𝑖)))
3418mptex 6527 . . . . . . . . . . . 12 (𝑖𝐴 ↦ (𝑒𝑖)) ∈ V
3526, 34eqeltri 2726 . . . . . . . . . . 11 𝐸 ∈ V
3635elixp 7957 . . . . . . . . . 10 (𝐸X𝑖𝐴 (𝑁𝑖) ↔ (𝐸 Fn 𝐴 ∧ ∀𝑖𝐴 (𝐸𝑖) ∈ (𝑁𝑖)))
3733, 36sylibr 224 . . . . . . . . 9 (∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → 𝐸X𝑖𝐴 (𝑁𝑖))
38 konigth.3 . . . . . . . . 9 𝑃 = X𝑖𝐴 (𝑁𝑖)
3937, 38syl6eleqr 2741 . . . . . . . 8 (∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → 𝐸𝑃)
40 foelrn 6418 . . . . . . . . . 10 ((𝑓:𝑆onto𝑃𝐸𝑃) → ∃𝑎𝑆 𝐸 = (𝑓𝑎))
4140expcom 450 . . . . . . . . 9 (𝐸𝑃 → (𝑓:𝑆onto𝑃 → ∃𝑎𝑆 𝐸 = (𝑓𝑎)))
42 konigth.2 . . . . . . . . . . . . . . 15 𝑆 = 𝑖𝐴 (𝑀𝑖)
4342eleq2i 2722 . . . . . . . . . . . . . 14 (𝑎𝑆𝑎 𝑖𝐴 (𝑀𝑖))
44 eliun 4556 . . . . . . . . . . . . . 14 (𝑎 𝑖𝐴 (𝑀𝑖) ↔ ∃𝑖𝐴 𝑎 ∈ (𝑀𝑖))
4543, 44bitri 264 . . . . . . . . . . . . 13 (𝑎𝑆 ↔ ∃𝑖𝐴 𝑎 ∈ (𝑀𝑖))
46 nfra1 2970 . . . . . . . . . . . . . . 15 𝑖𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖))
47 nfv 1883 . . . . . . . . . . . . . . 15 𝑖 𝐸 = (𝑓𝑎)
4846, 47nfan 1868 . . . . . . . . . . . . . 14 𝑖(∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) ∧ 𝐸 = (𝑓𝑎))
49 nfv 1883 . . . . . . . . . . . . . 14 𝑖 ¬ 𝑓 = 𝑓
5028ad2antrl 764 . . . . . . . . . . . . . . . . . . . 20 ((𝐸 = (𝑓𝑎) ∧ (𝑖𝐴𝑎 ∈ (𝑀𝑖))) → (𝐸𝑖) = (𝑒𝑖))
51 fveq1 6228 . . . . . . . . . . . . . . . . . . . . 21 (𝐸 = (𝑓𝑎) → (𝐸𝑖) = ((𝑓𝑎)‘𝑖))
528fveq1d 6231 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖𝐴 → ((𝐷𝑖)‘𝑎) = ((𝑎 ∈ (𝑀𝑖) ↦ ((𝑓𝑎)‘𝑖))‘𝑎))
533fvmpt2 6330 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑎 ∈ (𝑀𝑖) ∧ ((𝑓𝑎)‘𝑖) ∈ V) → ((𝑎 ∈ (𝑀𝑖) ↦ ((𝑓𝑎)‘𝑖))‘𝑎) = ((𝑓𝑎)‘𝑖))
542, 53mpan2 707 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 ∈ (𝑀𝑖) → ((𝑎 ∈ (𝑀𝑖) ↦ ((𝑓𝑎)‘𝑖))‘𝑎) = ((𝑓𝑎)‘𝑖))
5552, 54sylan9eq 2705 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖𝐴𝑎 ∈ (𝑀𝑖)) → ((𝐷𝑖)‘𝑎) = ((𝑓𝑎)‘𝑖))
5655eqcomd 2657 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖𝐴𝑎 ∈ (𝑀𝑖)) → ((𝑓𝑎)‘𝑖) = ((𝐷𝑖)‘𝑎))
5751, 56sylan9eq 2705 . . . . . . . . . . . . . . . . . . . 20 ((𝐸 = (𝑓𝑎) ∧ (𝑖𝐴𝑎 ∈ (𝑀𝑖))) → (𝐸𝑖) = ((𝐷𝑖)‘𝑎))
5850, 57eqtr3d 2687 . . . . . . . . . . . . . . . . . . 19 ((𝐸 = (𝑓𝑎) ∧ (𝑖𝐴𝑎 ∈ (𝑀𝑖))) → (𝑒𝑖) = ((𝐷𝑖)‘𝑎))
59 fnfvelrn 6396 . . . . . . . . . . . . . . . . . . . . 21 (((𝐷𝑖) Fn (𝑀𝑖) ∧ 𝑎 ∈ (𝑀𝑖)) → ((𝐷𝑖)‘𝑎) ∈ ran (𝐷𝑖))
6010, 59sylan 487 . . . . . . . . . . . . . . . . . . . 20 ((𝑖𝐴𝑎 ∈ (𝑀𝑖)) → ((𝐷𝑖)‘𝑎) ∈ ran (𝐷𝑖))
6160adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝐸 = (𝑓𝑎) ∧ (𝑖𝐴𝑎 ∈ (𝑀𝑖))) → ((𝐷𝑖)‘𝑎) ∈ ran (𝐷𝑖))
6258, 61eqeltrd 2730 . . . . . . . . . . . . . . . . . 18 ((𝐸 = (𝑓𝑎) ∧ (𝑖𝐴𝑎 ∈ (𝑀𝑖))) → (𝑒𝑖) ∈ ran (𝐷𝑖))
63623adant1 1099 . . . . . . . . . . . . . . . . 17 ((∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) ∧ 𝐸 = (𝑓𝑎) ∧ (𝑖𝐴𝑎 ∈ (𝑀𝑖))) → (𝑒𝑖) ∈ ran (𝐷𝑖))
64 simp1 1081 . . . . . . . . . . . . . . . . . 18 ((∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) ∧ 𝐸 = (𝑓𝑎) ∧ (𝑖𝐴𝑎 ∈ (𝑀𝑖))) → ∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)))
65 simp3l 1109 . . . . . . . . . . . . . . . . . 18 ((∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) ∧ 𝐸 = (𝑓𝑎) ∧ (𝑖𝐴𝑎 ∈ (𝑀𝑖))) → 𝑖𝐴)
66 rsp 2958 . . . . . . . . . . . . . . . . . . 19 (∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → (𝑖𝐴 → (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖))))
67 eldifn 3766 . . . . . . . . . . . . . . . . . . 19 ((𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → ¬ (𝑒𝑖) ∈ ran (𝐷𝑖))
6866, 67syl6 35 . . . . . . . . . . . . . . . . . 18 (∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → (𝑖𝐴 → ¬ (𝑒𝑖) ∈ ran (𝐷𝑖)))
6964, 65, 68sylc 65 . . . . . . . . . . . . . . . . 17 ((∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) ∧ 𝐸 = (𝑓𝑎) ∧ (𝑖𝐴𝑎 ∈ (𝑀𝑖))) → ¬ (𝑒𝑖) ∈ ran (𝐷𝑖))
7063, 69pm2.21dd 186 . . . . . . . . . . . . . . . 16 ((∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) ∧ 𝐸 = (𝑓𝑎) ∧ (𝑖𝐴𝑎 ∈ (𝑀𝑖))) → ¬ 𝑓 = 𝑓)
71703expia 1286 . . . . . . . . . . . . . . 15 ((∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) ∧ 𝐸 = (𝑓𝑎)) → ((𝑖𝐴𝑎 ∈ (𝑀𝑖)) → ¬ 𝑓 = 𝑓))
7271expd 451 . . . . . . . . . . . . . 14 ((∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) ∧ 𝐸 = (𝑓𝑎)) → (𝑖𝐴 → (𝑎 ∈ (𝑀𝑖) → ¬ 𝑓 = 𝑓)))
7348, 49, 72rexlimd 3055 . . . . . . . . . . . . 13 ((∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) ∧ 𝐸 = (𝑓𝑎)) → (∃𝑖𝐴 𝑎 ∈ (𝑀𝑖) → ¬ 𝑓 = 𝑓))
7445, 73syl5bi 232 . . . . . . . . . . . 12 ((∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) ∧ 𝐸 = (𝑓𝑎)) → (𝑎𝑆 → ¬ 𝑓 = 𝑓))
7574ex 449 . . . . . . . . . . 11 (∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → (𝐸 = (𝑓𝑎) → (𝑎𝑆 → ¬ 𝑓 = 𝑓)))
7675com23 86 . . . . . . . . . 10 (∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → (𝑎𝑆 → (𝐸 = (𝑓𝑎) → ¬ 𝑓 = 𝑓)))
7776rexlimdv 3059 . . . . . . . . 9 (∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → (∃𝑎𝑆 𝐸 = (𝑓𝑎) → ¬ 𝑓 = 𝑓))
7841, 77syl9r 78 . . . . . . . 8 (∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → (𝐸𝑃 → (𝑓:𝑆onto𝑃 → ¬ 𝑓 = 𝑓)))
7939, 78mpd 15 . . . . . . 7 (∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → (𝑓:𝑆onto𝑃 → ¬ 𝑓 = 𝑓))
8023, 79mt2i 132 . . . . . 6 (∀𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → ¬ 𝑓:𝑆onto𝑃)
8180exlimiv 1898 . . . . 5 (∃𝑒𝑖𝐴 (𝑒𝑖) ∈ ((𝑁𝑖) ∖ ran (𝐷𝑖)) → ¬ 𝑓:𝑆onto𝑃)
8217, 22, 813syl 18 . . . 4 (∀𝑖𝐴 (𝑀𝑖) ≺ (𝑁𝑖) → ¬ 𝑓:𝑆onto𝑃)
8382nexdv 1904 . . 3 (∀𝑖𝐴 (𝑀𝑖) ≺ (𝑁𝑖) → ¬ ∃𝑓 𝑓:𝑆onto𝑃)
8410dom 8131 . . . . . . . 8 ∅ ≼ (𝑀𝑖)
85 domsdomtr 8136 . . . . . . . 8 ((∅ ≼ (𝑀𝑖) ∧ (𝑀𝑖) ≺ (𝑁𝑖)) → ∅ ≺ (𝑁𝑖))
8684, 85mpan 706 . . . . . . 7 ((𝑀𝑖) ≺ (𝑁𝑖) → ∅ ≺ (𝑁𝑖))
87190sdom 8132 . . . . . . 7 (∅ ≺ (𝑁𝑖) ↔ (𝑁𝑖) ≠ ∅)
8886, 87sylib 208 . . . . . 6 ((𝑀𝑖) ≺ (𝑁𝑖) → (𝑁𝑖) ≠ ∅)
8988ralimi 2981 . . . . 5 (∀𝑖𝐴 (𝑀𝑖) ≺ (𝑁𝑖) → ∀𝑖𝐴 (𝑁𝑖) ≠ ∅)
9038neeq1i 2887 . . . . . 6 (𝑃 ≠ ∅ ↔ X𝑖𝐴 (𝑁𝑖) ≠ ∅)
9119rgenw 2953 . . . . . . . . 9 𝑖𝐴 (𝑁𝑖) ∈ V
92 ixpexg 7974 . . . . . . . . 9 (∀𝑖𝐴 (𝑁𝑖) ∈ V → X𝑖𝐴 (𝑁𝑖) ∈ V)
9391, 92ax-mp 5 . . . . . . . 8 X𝑖𝐴 (𝑁𝑖) ∈ V
9438, 93eqeltri 2726 . . . . . . 7 𝑃 ∈ V
95940sdom 8132 . . . . . 6 (∅ ≺ 𝑃𝑃 ≠ ∅)
9618, 19ac9 9343 . . . . . 6 (∀𝑖𝐴 (𝑁𝑖) ≠ ∅ ↔ X𝑖𝐴 (𝑁𝑖) ≠ ∅)
9790, 95, 963bitr4i 292 . . . . 5 (∅ ≺ 𝑃 ↔ ∀𝑖𝐴 (𝑁𝑖) ≠ ∅)
9889, 97sylibr 224 . . . 4 (∀𝑖𝐴 (𝑀𝑖) ≺ (𝑁𝑖) → ∅ ≺ 𝑃)
9918, 1iunex 7189 . . . . . . 7 𝑖𝐴 (𝑀𝑖) ∈ V
10042, 99eqeltri 2726 . . . . . 6 𝑆 ∈ V
101 domtri 9416 . . . . . 6 ((𝑃 ∈ V ∧ 𝑆 ∈ V) → (𝑃𝑆 ↔ ¬ 𝑆𝑃))
10294, 100, 101mp2an 708 . . . . 5 (𝑃𝑆 ↔ ¬ 𝑆𝑃)
103102biimpri 218 . . . 4 𝑆𝑃𝑃𝑆)
104 fodomr 8152 . . . 4 ((∅ ≺ 𝑃𝑃𝑆) → ∃𝑓 𝑓:𝑆onto𝑃)
10598, 103, 104syl2an 493 . . 3 ((∀𝑖𝐴 (𝑀𝑖) ≺ (𝑁𝑖) ∧ ¬ 𝑆𝑃) → ∃𝑓 𝑓:𝑆onto𝑃)
10683, 105mtand 692 . 2 (∀𝑖𝐴 (𝑀𝑖) ≺ (𝑁𝑖) → ¬ ¬ 𝑆𝑃)
107106notnotrd 128 1 (∀𝑖𝐴 (𝑀𝑖) ≺ (𝑁𝑖) → 𝑆𝑃)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523  ∃wex 1744   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941  ∃wrex 2942  Vcvv 3231   ∖ cdif 3604  ∅c0 3948  ∪ ciun 4552   class class class wbr 4685   ↦ cmpt 4762  ran crn 5144   Fn wfn 5921  –onto→wfo 5924  ‘cfv 5926  Xcixp 7950   ≼ cdom 7995   ≺ csdm 7996 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-ac2 9323 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-er 7787  df-map 7901  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-card 8803  df-acn 8806  df-ac 8977 This theorem is referenced by:  konigth  9429
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