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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  ontowfo 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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