MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfac12lem3 Structured version   Visualization version   GIF version

Theorem dfac12lem3 10142
Description: Lemma for dfac12 10146. (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 6903 . . . 4 (πΊβ€˜π΄) ∈ V
21rnex 7905 . . 3 ran (πΊβ€˜π΄) ∈ V
3 ssid 4003 . . . . 5 𝐴 βŠ† 𝐴
4 dfac12.1 . . . . . 6 (πœ‘ β†’ 𝐴 ∈ On)
5 sseq1 4006 . . . . . . . . 9 (π‘š = 𝑛 β†’ (π‘š βŠ† 𝐴 ↔ 𝑛 βŠ† 𝐴))
6 fveq2 6890 . . . . . . . . . . 11 (π‘š = 𝑛 β†’ (πΊβ€˜π‘š) = (πΊβ€˜π‘›))
7 f1eq1 6781 . . . . . . . . . . 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 6890 . . . . . . . . . . 11 (π‘š = 𝑛 β†’ (𝑅1β€˜π‘š) = (𝑅1β€˜π‘›))
10 f1eq2 6782 . . . . . . . . . . 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 278 . . . . . . . . 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 343 . . . . . . . 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 339 . . . . . . 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 4006 . . . . . . . . 9 (π‘š = 𝐴 β†’ (π‘š βŠ† 𝐴 ↔ 𝐴 βŠ† 𝐴))
16 fveq2 6890 . . . . . . . . . . 11 (π‘š = 𝐴 β†’ (πΊβ€˜π‘š) = (πΊβ€˜π΄))
17 f1eq1 6781 . . . . . . . . . . 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 6890 . . . . . . . . . . 11 (π‘š = 𝐴 β†’ (𝑅1β€˜π‘š) = (𝑅1β€˜π΄))
20 f1eq2 6782 . . . . . . . . . . 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 278 . . . . . . . . 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 343 . . . . . . . 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 339 . . . . . . 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 3177 . . . . . . . 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 6373 . . . . . . . . . . . . . . . . . 18 (π‘š ∈ On β†’ Ord π‘š)
2726ad2antrl 724 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ (π‘š ∈ On ∧ π‘š βŠ† 𝐴)) β†’ Ord π‘š)
28 ordelss 6379 . . . . . . . . . . . . . . . . 17 ((Ord π‘š ∧ 𝑛 ∈ π‘š) β†’ 𝑛 βŠ† π‘š)
2927, 28sylan 578 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (π‘š ∈ On ∧ π‘š βŠ† 𝐴)) ∧ 𝑛 ∈ π‘š) β†’ 𝑛 βŠ† π‘š)
30 simplrr 774 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (π‘š ∈ On ∧ π‘š βŠ† 𝐴)) ∧ 𝑛 ∈ π‘š) β†’ π‘š βŠ† 𝐴)
3129, 30sstrd 3991 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (π‘š ∈ On ∧ π‘š βŠ† 𝐴)) ∧ 𝑛 ∈ π‘š) β†’ 𝑛 βŠ† 𝐴)
32 pm5.5 360 . . . . . . . . . . . . . . 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 3173 . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . 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 773 . . . . . . . . . . . . . . 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 2730 . . . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . . 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 6890 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑧 β†’ (πΊβ€˜π‘›) = (πΊβ€˜π‘§))
44 f1eq1 6781 . . . . . . . . . . . . . . . . . . 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 6890 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑧 β†’ (𝑅1β€˜π‘›) = (𝑅1β€˜π‘§))
47 f1eq2 6782 . . . . . . . . . . . . . . . . . . 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 278 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑧 β†’ ((πΊβ€˜π‘›):(𝑅1β€˜π‘›)–1-1β†’On ↔ (πΊβ€˜π‘§):(𝑅1β€˜π‘§)–1-1β†’On))
5049cbvralvw 3232 . . . . . . . . . . . . . . . 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 10141 . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . 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 455 . . . . . . . . . . 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 412 . . . . . . . . 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 7849 . . . . . 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 6786 . . . 4 ((πΊβ€˜π΄):(𝑅1β€˜π΄)–1-1β†’On β†’ (πΊβ€˜π΄):(𝑅1β€˜π΄)⟢On)
64 frn 6723 . . . 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 10037 . . 3 ((ran (πΊβ€˜π΄) ∈ V ∧ ran (πΊβ€˜π΄) βŠ† On) β†’ ran (πΊβ€˜π΄) ∈ dom card)
672, 65, 66sylancr 585 . 2 (πœ‘ β†’ ran (πΊβ€˜π΄) ∈ dom card)
68 f1f1orn 6843 . . . 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 6903 . . . 4 (𝑅1β€˜π΄) ∈ V
7170f1oen 8971 . . 3 ((πΊβ€˜π΄):(𝑅1β€˜π΄)–1-1-ontoβ†’ran (πΊβ€˜π΄) β†’ (𝑅1β€˜π΄) β‰ˆ ran (πΊβ€˜π΄))
72 ennum 9944 . . 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 256 1 (πœ‘ β†’ (𝑅1β€˜π΄) ∈ dom card)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  Vcvv 3472   βŠ† wss 3947  ifcif 4527  π’« cpw 4601  βˆͺ cuni 4907   class class class wbr 5147   ↦ cmpt 5230   E cep 5578  β—‘ccnv 5674  dom cdm 5675  ran crn 5676   β€œ cima 5678   ∘ ccom 5679  Ord word 6362  Oncon0 6363  suc csuc 6365  βŸΆwf 6538  β€“1-1β†’wf1 6539  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542  (class class class)co 7411  recscrecs 8372   +o coa 8465   Β·o comu 8466   β‰ˆ cen 8938  OrdIsocoi 9506  harchar 9553  π‘…1cr1 9759  rankcrnk 9760  cardccrd 9932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-oadd 8472  df-omul 8473  df-er 8705  df-en 8942  df-dom 8943  df-oi 9507  df-har 9554  df-r1 9761  df-rank 9762  df-card 9936
This theorem is referenced by:  dfac12r  10143
  Copyright terms: Public domain W3C validator