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Theorem dfac12lem3 10089
Description: Lemma for dfac12 10093. (Contributed by Mario Carneiro, 29-May-2015.)
Hypotheses
Ref Expression
dfac12.1 (πœ‘ β†’ 𝐴 ∈ On)
dfac12.3 (πœ‘ β†’ 𝐹:𝒫 (harβ€˜(𝑅1β€˜π΄))–1-1β†’On)
dfac12.4 𝐺 = recs((π‘₯ ∈ V ↦ (𝑦 ∈ (𝑅1β€˜dom π‘₯) ↦ if(dom π‘₯ = βˆͺ dom π‘₯, ((suc βˆͺ ran βˆͺ ran π‘₯ Β·o (rankβ€˜π‘¦)) +o ((π‘₯β€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜((β—‘OrdIso( E , ran (π‘₯β€˜βˆͺ dom π‘₯)) ∘ (π‘₯β€˜βˆͺ dom π‘₯)) β€œ 𝑦))))))
Assertion
Ref Expression
dfac12lem3 (πœ‘ β†’ (𝑅1β€˜π΄) ∈ dom card)
Distinct variable groups:   𝑦,𝐴   π‘₯,𝑦,𝐺   πœ‘,𝑦   π‘₯,𝐹,𝑦
Allowed substitution hints:   πœ‘(π‘₯)   𝐴(π‘₯)

Proof of Theorem dfac12lem3
Dummy variables π‘š 𝑛 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6859 . . . 4 (πΊβ€˜π΄) ∈ V
21rnex 7853 . . 3 ran (πΊβ€˜π΄) ∈ V
3 ssid 3970 . . . . 5 𝐴 βŠ† 𝐴
4 dfac12.1 . . . . . 6 (πœ‘ β†’ 𝐴 ∈ On)
5 sseq1 3973 . . . . . . . . 9 (π‘š = 𝑛 β†’ (π‘š βŠ† 𝐴 ↔ 𝑛 βŠ† 𝐴))
6 fveq2 6846 . . . . . . . . . . 11 (π‘š = 𝑛 β†’ (πΊβ€˜π‘š) = (πΊβ€˜π‘›))
7 f1eq1 6737 . . . . . . . . . . 11 ((πΊβ€˜π‘š) = (πΊβ€˜π‘›) β†’ ((πΊβ€˜π‘š):(𝑅1β€˜π‘š)–1-1β†’On ↔ (πΊβ€˜π‘›):(𝑅1β€˜π‘š)–1-1β†’On))
86, 7syl 17 . . . . . . . . . 10 (π‘š = 𝑛 β†’ ((πΊβ€˜π‘š):(𝑅1β€˜π‘š)–1-1β†’On ↔ (πΊβ€˜π‘›):(𝑅1β€˜π‘š)–1-1β†’On))
9 fveq2 6846 . . . . . . . . . . 11 (π‘š = 𝑛 β†’ (𝑅1β€˜π‘š) = (𝑅1β€˜π‘›))
10 f1eq2 6738 . . . . . . . . . . 11 ((𝑅1β€˜π‘š) = (𝑅1β€˜π‘›) β†’ ((πΊβ€˜π‘›):(𝑅1β€˜π‘š)–1-1β†’On ↔ (πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On))
119, 10syl 17 . . . . . . . . . 10 (π‘š = 𝑛 β†’ ((πΊβ€˜π‘›):(𝑅1β€˜π‘š)–1-1β†’On ↔ (πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On))
128, 11bitrd 279 . . . . . . . . 9 (π‘š = 𝑛 β†’ ((πΊβ€˜π‘š):(𝑅1β€˜π‘š)–1-1β†’On ↔ (πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On))
135, 12imbi12d 345 . . . . . . . 8 (π‘š = 𝑛 β†’ ((π‘š βŠ† 𝐴 β†’ (πΊβ€˜π‘š):(𝑅1β€˜π‘š)–1-1β†’On) ↔ (𝑛 βŠ† 𝐴 β†’ (πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On)))
1413imbi2d 341 . . . . . . 7 (π‘š = 𝑛 β†’ ((πœ‘ β†’ (π‘š βŠ† 𝐴 β†’ (πΊβ€˜π‘š):(𝑅1β€˜π‘š)–1-1β†’On)) ↔ (πœ‘ β†’ (𝑛 βŠ† 𝐴 β†’ (πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On))))
15 sseq1 3973 . . . . . . . . 9 (π‘š = 𝐴 β†’ (π‘š βŠ† 𝐴 ↔ 𝐴 βŠ† 𝐴))
16 fveq2 6846 . . . . . . . . . . 11 (π‘š = 𝐴 β†’ (πΊβ€˜π‘š) = (πΊβ€˜π΄))
17 f1eq1 6737 . . . . . . . . . . 11 ((πΊβ€˜π‘š) = (πΊβ€˜π΄) β†’ ((πΊβ€˜π‘š):(𝑅1β€˜π‘š)–1-1β†’On ↔ (πΊβ€˜π΄):(𝑅1β€˜π‘š)–1-1β†’On))
1816, 17syl 17 . . . . . . . . . 10 (π‘š = 𝐴 β†’ ((πΊβ€˜π‘š):(𝑅1β€˜π‘š)–1-1β†’On ↔ (πΊβ€˜π΄):(𝑅1β€˜π‘š)–1-1β†’On))
19 fveq2 6846 . . . . . . . . . . 11 (π‘š = 𝐴 β†’ (𝑅1β€˜π‘š) = (𝑅1β€˜π΄))
20 f1eq2 6738 . . . . . . . . . . 11 ((𝑅1β€˜π‘š) = (𝑅1β€˜π΄) β†’ ((πΊβ€˜π΄):(𝑅1β€˜π‘š)–1-1β†’On ↔ (πΊβ€˜π΄):(𝑅1β€˜π΄)–1-1β†’On))
2119, 20syl 17 . . . . . . . . . 10 (π‘š = 𝐴 β†’ ((πΊβ€˜π΄):(𝑅1β€˜π‘š)–1-1β†’On ↔ (πΊβ€˜π΄):(𝑅1β€˜π΄)–1-1β†’On))
2218, 21bitrd 279 . . . . . . . . 9 (π‘š = 𝐴 β†’ ((πΊβ€˜π‘š):(𝑅1β€˜π‘š)–1-1β†’On ↔ (πΊβ€˜π΄):(𝑅1β€˜π΄)–1-1β†’On))
2315, 22imbi12d 345 . . . . . . . 8 (π‘š = 𝐴 β†’ ((π‘š βŠ† 𝐴 β†’ (πΊβ€˜π‘š):(𝑅1β€˜π‘š)–1-1β†’On) ↔ (𝐴 βŠ† 𝐴 β†’ (πΊβ€˜π΄):(𝑅1β€˜π΄)–1-1β†’On)))
2423imbi2d 341 . . . . . . 7 (π‘š = 𝐴 β†’ ((πœ‘ β†’ (π‘š βŠ† 𝐴 β†’ (πΊβ€˜π‘š):(𝑅1β€˜π‘š)–1-1β†’On)) ↔ (πœ‘ β†’ (𝐴 βŠ† 𝐴 β†’ (πΊβ€˜π΄):(𝑅1β€˜π΄)–1-1β†’On))))
25 r19.21v 3173 . . . . . . . 8 (βˆ€π‘› ∈ π‘š (πœ‘ β†’ (𝑛 βŠ† 𝐴 β†’ (πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On)) ↔ (πœ‘ β†’ βˆ€π‘› ∈ π‘š (𝑛 βŠ† 𝐴 β†’ (πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On)))
26 eloni 6331 . . . . . . . . . . . . . . . . . 18 (π‘š ∈ On β†’ Ord π‘š)
2726ad2antrl 727 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ (π‘š ∈ On ∧ π‘š βŠ† 𝐴)) β†’ Ord π‘š)
28 ordelss 6337 . . . . . . . . . . . . . . . . 17 ((Ord π‘š ∧ 𝑛 ∈ π‘š) β†’ 𝑛 βŠ† π‘š)
2927, 28sylan 581 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (π‘š ∈ On ∧ π‘š βŠ† 𝐴)) ∧ 𝑛 ∈ π‘š) β†’ 𝑛 βŠ† π‘š)
30 simplrr 777 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (π‘š ∈ On ∧ π‘š βŠ† 𝐴)) ∧ 𝑛 ∈ π‘š) β†’ π‘š βŠ† 𝐴)
3129, 30sstrd 3958 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (π‘š ∈ On ∧ π‘š βŠ† 𝐴)) ∧ 𝑛 ∈ π‘š) β†’ 𝑛 βŠ† 𝐴)
32 pm5.5 362 . . . . . . . . . . . . . . 15 (𝑛 βŠ† 𝐴 β†’ ((𝑛 βŠ† 𝐴 β†’ (πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On) ↔ (πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On))
3331, 32syl 17 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (π‘š ∈ On ∧ π‘š βŠ† 𝐴)) ∧ 𝑛 ∈ π‘š) β†’ ((𝑛 βŠ† 𝐴 β†’ (πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On) ↔ (πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On))
3433ralbidva 3169 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘š ∈ On ∧ π‘š βŠ† 𝐴)) β†’ (βˆ€π‘› ∈ π‘š (𝑛 βŠ† 𝐴 β†’ (πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On) ↔ βˆ€π‘› ∈ π‘š (πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On))
354ad2antrr 725 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (π‘š ∈ On ∧ π‘š βŠ† 𝐴)) ∧ βˆ€π‘› ∈ π‘š (πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On) β†’ 𝐴 ∈ On)
36 dfac12.3 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ 𝐹:𝒫 (harβ€˜(𝑅1β€˜π΄))–1-1β†’On)
3736ad2antrr 725 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (π‘š ∈ On ∧ π‘š βŠ† 𝐴)) ∧ βˆ€π‘› ∈ π‘š (πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On) β†’ 𝐹:𝒫 (harβ€˜(𝑅1β€˜π΄))–1-1β†’On)
38 dfac12.4 . . . . . . . . . . . . . . 15 𝐺 = recs((π‘₯ ∈ V ↦ (𝑦 ∈ (𝑅1β€˜dom π‘₯) ↦ if(dom π‘₯ = βˆͺ dom π‘₯, ((suc βˆͺ ran βˆͺ ran π‘₯ Β·o (rankβ€˜π‘¦)) +o ((π‘₯β€˜suc (rankβ€˜π‘¦))β€˜π‘¦)), (πΉβ€˜((β—‘OrdIso( E , ran (π‘₯β€˜βˆͺ dom π‘₯)) ∘ (π‘₯β€˜βˆͺ dom π‘₯)) β€œ 𝑦))))))
39 simplrl 776 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (π‘š ∈ On ∧ π‘š βŠ† 𝐴)) ∧ βˆ€π‘› ∈ π‘š (πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On) β†’ π‘š ∈ On)
40 eqid 2733 . . . . . . . . . . . . . . 15 (β—‘OrdIso( E , ran (πΊβ€˜βˆͺ π‘š)) ∘ (πΊβ€˜βˆͺ π‘š)) = (β—‘OrdIso( E , ran (πΊβ€˜βˆͺ π‘š)) ∘ (πΊβ€˜βˆͺ π‘š))
41 simplrr 777 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (π‘š ∈ On ∧ π‘š βŠ† 𝐴)) ∧ βˆ€π‘› ∈ π‘š (πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On) β†’ π‘š βŠ† 𝐴)
42 simpr 486 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (π‘š ∈ On ∧ π‘š βŠ† 𝐴)) ∧ βˆ€π‘› ∈ π‘š (πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On) β†’ βˆ€π‘› ∈ π‘š (πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On)
43 fveq2 6846 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑧 β†’ (πΊβ€˜π‘›) = (πΊβ€˜π‘§))
44 f1eq1 6737 . . . . . . . . . . . . . . . . . . 19 ((πΊβ€˜π‘›) = (πΊβ€˜π‘§) β†’ ((πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On ↔ (πΊβ€˜π‘§):(𝑅1β€˜π‘›)–1-1β†’On))
4543, 44syl 17 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑧 β†’ ((πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On ↔ (πΊβ€˜π‘§):(𝑅1β€˜π‘›)–1-1β†’On))
46 fveq2 6846 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑧 β†’ (𝑅1β€˜π‘›) = (𝑅1β€˜π‘§))
47 f1eq2 6738 . . . . . . . . . . . . . . . . . . 19 ((𝑅1β€˜π‘›) = (𝑅1β€˜π‘§) β†’ ((πΊβ€˜π‘§):(𝑅1β€˜π‘›)–1-1β†’On ↔ (πΊβ€˜π‘§):(𝑅1β€˜π‘§)–1-1β†’On))
4846, 47syl 17 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑧 β†’ ((πΊβ€˜π‘§):(𝑅1β€˜π‘›)–1-1β†’On ↔ (πΊβ€˜π‘§):(𝑅1β€˜π‘§)–1-1β†’On))
4945, 48bitrd 279 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑧 β†’ ((πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On ↔ (πΊβ€˜π‘§):(𝑅1β€˜π‘§)–1-1β†’On))
5049cbvralvw 3224 . . . . . . . . . . . . . . . 16 (βˆ€π‘› ∈ π‘š (πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On ↔ βˆ€π‘§ ∈ π‘š (πΊβ€˜π‘§):(𝑅1β€˜π‘§)–1-1β†’On)
5142, 50sylib 217 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (π‘š ∈ On ∧ π‘š βŠ† 𝐴)) ∧ βˆ€π‘› ∈ π‘š (πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On) β†’ βˆ€π‘§ ∈ π‘š (πΊβ€˜π‘§):(𝑅1β€˜π‘§)–1-1β†’On)
5235, 37, 38, 39, 40, 41, 51dfac12lem2 10088 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (π‘š ∈ On ∧ π‘š βŠ† 𝐴)) ∧ βˆ€π‘› ∈ π‘š (πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On) β†’ (πΊβ€˜π‘š):(𝑅1β€˜π‘š)–1-1β†’On)
5352ex 414 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘š ∈ On ∧ π‘š βŠ† 𝐴)) β†’ (βˆ€π‘› ∈ π‘š (πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On β†’ (πΊβ€˜π‘š):(𝑅1β€˜π‘š)–1-1β†’On))
5434, 53sylbid 239 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘š ∈ On ∧ π‘š βŠ† 𝐴)) β†’ (βˆ€π‘› ∈ π‘š (𝑛 βŠ† 𝐴 β†’ (πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On) β†’ (πΊβ€˜π‘š):(𝑅1β€˜π‘š)–1-1β†’On))
5554expr 458 . . . . . . . . . . 11 ((πœ‘ ∧ π‘š ∈ On) β†’ (π‘š βŠ† 𝐴 β†’ (βˆ€π‘› ∈ π‘š (𝑛 βŠ† 𝐴 β†’ (πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On) β†’ (πΊβ€˜π‘š):(𝑅1β€˜π‘š)–1-1β†’On)))
5655com23 86 . . . . . . . . . 10 ((πœ‘ ∧ π‘š ∈ On) β†’ (βˆ€π‘› ∈ π‘š (𝑛 βŠ† 𝐴 β†’ (πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On) β†’ (π‘š βŠ† 𝐴 β†’ (πΊβ€˜π‘š):(𝑅1β€˜π‘š)–1-1β†’On)))
5756expcom 415 . . . . . . . . 9 (π‘š ∈ On β†’ (πœ‘ β†’ (βˆ€π‘› ∈ π‘š (𝑛 βŠ† 𝐴 β†’ (πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On) β†’ (π‘š βŠ† 𝐴 β†’ (πΊβ€˜π‘š):(𝑅1β€˜π‘š)–1-1β†’On))))
5857a2d 29 . . . . . . . 8 (π‘š ∈ On β†’ ((πœ‘ β†’ βˆ€π‘› ∈ π‘š (𝑛 βŠ† 𝐴 β†’ (πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On)) β†’ (πœ‘ β†’ (π‘š βŠ† 𝐴 β†’ (πΊβ€˜π‘š):(𝑅1β€˜π‘š)–1-1β†’On))))
5925, 58biimtrid 241 . . . . . . 7 (π‘š ∈ On β†’ (βˆ€π‘› ∈ π‘š (πœ‘ β†’ (𝑛 βŠ† 𝐴 β†’ (πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On)) β†’ (πœ‘ β†’ (π‘š βŠ† 𝐴 β†’ (πΊβ€˜π‘š):(𝑅1β€˜π‘š)–1-1β†’On))))
6014, 24, 59tfis3 7798 . . . . . 6 (𝐴 ∈ On β†’ (πœ‘ β†’ (𝐴 βŠ† 𝐴 β†’ (πΊβ€˜π΄):(𝑅1β€˜π΄)–1-1β†’On)))
614, 60mpcom 38 . . . . 5 (πœ‘ β†’ (𝐴 βŠ† 𝐴 β†’ (πΊβ€˜π΄):(𝑅1β€˜π΄)–1-1β†’On))
623, 61mpi 20 . . . 4 (πœ‘ β†’ (πΊβ€˜π΄):(𝑅1β€˜π΄)–1-1β†’On)
63 f1f 6742 . . . 4 ((πΊβ€˜π΄):(𝑅1β€˜π΄)–1-1β†’On β†’ (πΊβ€˜π΄):(𝑅1β€˜π΄)⟢On)
64 frn 6679 . . . 4 ((πΊβ€˜π΄):(𝑅1β€˜π΄)⟢On β†’ ran (πΊβ€˜π΄) βŠ† On)
6562, 63, 643syl 18 . . 3 (πœ‘ β†’ ran (πΊβ€˜π΄) βŠ† On)
66 onssnum 9984 . . 3 ((ran (πΊβ€˜π΄) ∈ V ∧ ran (πΊβ€˜π΄) βŠ† On) β†’ ran (πΊβ€˜π΄) ∈ dom card)
672, 65, 66sylancr 588 . 2 (πœ‘ β†’ ran (πΊβ€˜π΄) ∈ dom card)
68 f1f1orn 6799 . . . 4 ((πΊβ€˜π΄):(𝑅1β€˜π΄)–1-1β†’On β†’ (πΊβ€˜π΄):(𝑅1β€˜π΄)–1-1-ontoβ†’ran (πΊβ€˜π΄))
6962, 68syl 17 . . 3 (πœ‘ β†’ (πΊβ€˜π΄):(𝑅1β€˜π΄)–1-1-ontoβ†’ran (πΊβ€˜π΄))
70 fvex 6859 . . . 4 (𝑅1β€˜π΄) ∈ V
7170f1oen 8919 . . 3 ((πΊβ€˜π΄):(𝑅1β€˜π΄)–1-1-ontoβ†’ran (πΊβ€˜π΄) β†’ (𝑅1β€˜π΄) β‰ˆ ran (πΊβ€˜π΄))
72 ennum 9891 . . 3 ((𝑅1β€˜π΄) β‰ˆ ran (πΊβ€˜π΄) β†’ ((𝑅1β€˜π΄) ∈ dom card ↔ ran (πΊβ€˜π΄) ∈ dom card))
7369, 71, 723syl 18 . 2 (πœ‘ β†’ ((𝑅1β€˜π΄) ∈ dom card ↔ ran (πΊβ€˜π΄) ∈ dom card))
7467, 73mpbird 257 1 (πœ‘ β†’ (𝑅1β€˜π΄) ∈ dom card)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  Vcvv 3447   βŠ† wss 3914  ifcif 4490  π’« cpw 4564  βˆͺ cuni 4869   class class class wbr 5109   ↦ cmpt 5192   E cep 5540  β—‘ccnv 5636  dom cdm 5637  ran crn 5638   β€œ cima 5640   ∘ ccom 5641  Ord word 6320  Oncon0 6321  suc csuc 6323  βŸΆwf 6496  β€“1-1β†’wf1 6497  β€“1-1-ontoβ†’wf1o 6499  β€˜cfv 6500  (class class class)co 7361  recscrecs 8320   +o coa 8413   Β·o comu 8414   β‰ˆ cen 8886  OrdIsocoi 9453  harchar 9500  π‘…1cr1 9706  rankcrnk 9707  cardccrd 9879
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-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  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 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-se 5593  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-oadd 8420  df-omul 8421  df-er 8654  df-en 8890  df-dom 8891  df-oi 9454  df-har 9501  df-r1 9708  df-rank 9709  df-card 9883
This theorem is referenced by:  dfac12r  10090
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