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Theorem cfflb 10251
Description: If there is a cofinal map from 𝐡 to 𝐴, then 𝐡 is at least (cfβ€˜π΄). This theorem and cff1 10250 motivate the picture of (cfβ€˜π΄) as the greatest lower bound of the domain of cofinal maps into 𝐴. (Contributed by Mario Carneiro, 28-Feb-2013.)
Assertion
Ref Expression
cfflb ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (βˆƒπ‘“(𝑓:𝐡⟢𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝐡 𝑧 βŠ† (π‘“β€˜π‘€)) β†’ (cfβ€˜π΄) βŠ† 𝐡))
Distinct variable groups:   𝐴,𝑓,𝑀,𝑧   𝐡,𝑓,𝑀,𝑧
Proof of Theorem cfflb
Dummy variables 𝑠 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frn 6722 . . . . . . 7 (𝑓:𝐡⟢𝐴 β†’ ran 𝑓 βŠ† 𝐴)
21adantr 482 . . . . . 6 ((𝑓:𝐡⟢𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝐡 𝑧 βŠ† (π‘“β€˜π‘€)) β†’ ran 𝑓 βŠ† 𝐴)
3 ffn 6715 . . . . . . . . . . 11 (𝑓:𝐡⟢𝐴 β†’ 𝑓 Fn 𝐡)
4 fnfvelrn 7080 . . . . . . . . . . 11 ((𝑓 Fn 𝐡 ∧ 𝑀 ∈ 𝐡) β†’ (π‘“β€˜π‘€) ∈ ran 𝑓)
53, 4sylan 581 . . . . . . . . . 10 ((𝑓:𝐡⟢𝐴 ∧ 𝑀 ∈ 𝐡) β†’ (π‘“β€˜π‘€) ∈ ran 𝑓)
6 sseq2 4008 . . . . . . . . . . 11 (𝑠 = (π‘“β€˜π‘€) β†’ (𝑧 βŠ† 𝑠 ↔ 𝑧 βŠ† (π‘“β€˜π‘€)))
76rspcev 3613 . . . . . . . . . 10 (((π‘“β€˜π‘€) ∈ ran 𝑓 ∧ 𝑧 βŠ† (π‘“β€˜π‘€)) β†’ βˆƒπ‘  ∈ ran 𝑓 𝑧 βŠ† 𝑠)
85, 7sylan 581 . . . . . . . . 9 (((𝑓:𝐡⟢𝐴 ∧ 𝑀 ∈ 𝐡) ∧ 𝑧 βŠ† (π‘“β€˜π‘€)) β†’ βˆƒπ‘  ∈ ran 𝑓 𝑧 βŠ† 𝑠)
98rexlimdva2 3158 . . . . . . . 8 (𝑓:𝐡⟢𝐴 β†’ (βˆƒπ‘€ ∈ 𝐡 𝑧 βŠ† (π‘“β€˜π‘€) β†’ βˆƒπ‘  ∈ ran 𝑓 𝑧 βŠ† 𝑠))
109ralimdv 3170 . . . . . . 7 (𝑓:𝐡⟢𝐴 β†’ (βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝐡 𝑧 βŠ† (π‘“β€˜π‘€) β†’ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘  ∈ ran 𝑓 𝑧 βŠ† 𝑠))
1110imp 408 . . . . . 6 ((𝑓:𝐡⟢𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝐡 𝑧 βŠ† (π‘“β€˜π‘€)) β†’ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘  ∈ ran 𝑓 𝑧 βŠ† 𝑠)
122, 11jca 513 . . . . 5 ((𝑓:𝐡⟢𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝐡 𝑧 βŠ† (π‘“β€˜π‘€)) β†’ (ran 𝑓 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘  ∈ ran 𝑓 𝑧 βŠ† 𝑠))
13 fvex 6902 . . . . . 6 (cardβ€˜ran 𝑓) ∈ V
14 cfval 10239 . . . . . . . . . . 11 (𝐴 ∈ On β†’ (cfβ€˜π΄) = ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘  ∈ 𝑦 𝑧 βŠ† 𝑠))})
1514adantr 482 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (cfβ€˜π΄) = ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘  ∈ 𝑦 𝑧 βŠ† 𝑠))})
16153ad2ant2 1135 . . . . . . . . 9 ((π‘₯ = (cardβ€˜ran 𝑓) ∧ (𝐴 ∈ On ∧ 𝐡 ∈ On) ∧ (ran 𝑓 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘  ∈ ran 𝑓 𝑧 βŠ† 𝑠)) β†’ (cfβ€˜π΄) = ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘  ∈ 𝑦 𝑧 βŠ† 𝑠))})
17 vex 3479 . . . . . . . . . . . . . 14 𝑓 ∈ V
1817rnex 7900 . . . . . . . . . . . . 13 ran 𝑓 ∈ V
19 fveq2 6889 . . . . . . . . . . . . . . 15 (𝑦 = ran 𝑓 β†’ (cardβ€˜π‘¦) = (cardβ€˜ran 𝑓))
2019eqeq2d 2744 . . . . . . . . . . . . . 14 (𝑦 = ran 𝑓 β†’ (π‘₯ = (cardβ€˜π‘¦) ↔ π‘₯ = (cardβ€˜ran 𝑓)))
21 sseq1 4007 . . . . . . . . . . . . . . 15 (𝑦 = ran 𝑓 β†’ (𝑦 βŠ† 𝐴 ↔ ran 𝑓 βŠ† 𝐴))
22 rexeq 3322 . . . . . . . . . . . . . . . 16 (𝑦 = ran 𝑓 β†’ (βˆƒπ‘  ∈ 𝑦 𝑧 βŠ† 𝑠 ↔ βˆƒπ‘  ∈ ran 𝑓 𝑧 βŠ† 𝑠))
2322ralbidv 3178 . . . . . . . . . . . . . . 15 (𝑦 = ran 𝑓 β†’ (βˆ€π‘§ ∈ 𝐴 βˆƒπ‘  ∈ 𝑦 𝑧 βŠ† 𝑠 ↔ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘  ∈ ran 𝑓 𝑧 βŠ† 𝑠))
2421, 23anbi12d 632 . . . . . . . . . . . . . 14 (𝑦 = ran 𝑓 β†’ ((𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘  ∈ 𝑦 𝑧 βŠ† 𝑠) ↔ (ran 𝑓 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘  ∈ ran 𝑓 𝑧 βŠ† 𝑠)))
2520, 24anbi12d 632 . . . . . . . . . . . . 13 (𝑦 = ran 𝑓 β†’ ((π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘  ∈ 𝑦 𝑧 βŠ† 𝑠)) ↔ (π‘₯ = (cardβ€˜ran 𝑓) ∧ (ran 𝑓 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘  ∈ ran 𝑓 𝑧 βŠ† 𝑠))))
2618, 25spcev 3597 . . . . . . . . . . . 12 ((π‘₯ = (cardβ€˜ran 𝑓) ∧ (ran 𝑓 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘  ∈ ran 𝑓 𝑧 βŠ† 𝑠)) β†’ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘  ∈ 𝑦 𝑧 βŠ† 𝑠)))
27 abid 2714 . . . . . . . . . . . 12 (π‘₯ ∈ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘  ∈ 𝑦 𝑧 βŠ† 𝑠))} ↔ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘  ∈ 𝑦 𝑧 βŠ† 𝑠)))
2826, 27sylibr 233 . . . . . . . . . . 11 ((π‘₯ = (cardβ€˜ran 𝑓) ∧ (ran 𝑓 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘  ∈ ran 𝑓 𝑧 βŠ† 𝑠)) β†’ π‘₯ ∈ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘  ∈ 𝑦 𝑧 βŠ† 𝑠))})
29 intss1 4967 . . . . . . . . . . 11 (π‘₯ ∈ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘  ∈ 𝑦 𝑧 βŠ† 𝑠))} β†’ ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘  ∈ 𝑦 𝑧 βŠ† 𝑠))} βŠ† π‘₯)
3028, 29syl 17 . . . . . . . . . 10 ((π‘₯ = (cardβ€˜ran 𝑓) ∧ (ran 𝑓 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘  ∈ ran 𝑓 𝑧 βŠ† 𝑠)) β†’ ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘  ∈ 𝑦 𝑧 βŠ† 𝑠))} βŠ† π‘₯)
31303adant2 1132 . . . . . . . . 9 ((π‘₯ = (cardβ€˜ran 𝑓) ∧ (𝐴 ∈ On ∧ 𝐡 ∈ On) ∧ (ran 𝑓 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘  ∈ ran 𝑓 𝑧 βŠ† 𝑠)) β†’ ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘  ∈ 𝑦 𝑧 βŠ† 𝑠))} βŠ† π‘₯)
3216, 31eqsstrd 4020 . . . . . . . 8 ((π‘₯ = (cardβ€˜ran 𝑓) ∧ (𝐴 ∈ On ∧ 𝐡 ∈ On) ∧ (ran 𝑓 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘  ∈ ran 𝑓 𝑧 βŠ† 𝑠)) β†’ (cfβ€˜π΄) βŠ† π‘₯)
33323expib 1123 . . . . . . 7 (π‘₯ = (cardβ€˜ran 𝑓) β†’ (((𝐴 ∈ On ∧ 𝐡 ∈ On) ∧ (ran 𝑓 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘  ∈ ran 𝑓 𝑧 βŠ† 𝑠)) β†’ (cfβ€˜π΄) βŠ† π‘₯))
34 sseq2 4008 . . . . . . 7 (π‘₯ = (cardβ€˜ran 𝑓) β†’ ((cfβ€˜π΄) βŠ† π‘₯ ↔ (cfβ€˜π΄) βŠ† (cardβ€˜ran 𝑓)))
3533, 34sylibd 238 . . . . . 6 (π‘₯ = (cardβ€˜ran 𝑓) β†’ (((𝐴 ∈ On ∧ 𝐡 ∈ On) ∧ (ran 𝑓 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘  ∈ ran 𝑓 𝑧 βŠ† 𝑠)) β†’ (cfβ€˜π΄) βŠ† (cardβ€˜ran 𝑓)))
3613, 35vtocle 3576 . . . . 5 (((𝐴 ∈ On ∧ 𝐡 ∈ On) ∧ (ran 𝑓 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘  ∈ ran 𝑓 𝑧 βŠ† 𝑠)) β†’ (cfβ€˜π΄) βŠ† (cardβ€˜ran 𝑓))
3712, 36sylan2 594 . . . 4 (((𝐴 ∈ On ∧ 𝐡 ∈ On) ∧ (𝑓:𝐡⟢𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝐡 𝑧 βŠ† (π‘“β€˜π‘€))) β†’ (cfβ€˜π΄) βŠ† (cardβ€˜ran 𝑓))
38 cardidm 9951 . . . . . . 7 (cardβ€˜(cardβ€˜ran 𝑓)) = (cardβ€˜ran 𝑓)
39 onss 7769 . . . . . . . . . . . . . 14 (𝐴 ∈ On β†’ 𝐴 βŠ† On)
401, 39sylan9ssr 3996 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝑓:𝐡⟢𝐴) β†’ ran 𝑓 βŠ† On)
41403adant2 1132 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐡 ∈ On ∧ 𝑓:𝐡⟢𝐴) β†’ ran 𝑓 βŠ† On)
42 onssnum 10032 . . . . . . . . . . . 12 ((ran 𝑓 ∈ V ∧ ran 𝑓 βŠ† On) β†’ ran 𝑓 ∈ dom card)
4318, 41, 42sylancr 588 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝐡 ∈ On ∧ 𝑓:𝐡⟢𝐴) β†’ ran 𝑓 ∈ dom card)
44 cardid2 9945 . . . . . . . . . . 11 (ran 𝑓 ∈ dom card β†’ (cardβ€˜ran 𝑓) β‰ˆ ran 𝑓)
4543, 44syl 17 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐡 ∈ On ∧ 𝑓:𝐡⟢𝐴) β†’ (cardβ€˜ran 𝑓) β‰ˆ ran 𝑓)
46 onenon 9941 . . . . . . . . . . . . 13 (𝐡 ∈ On β†’ 𝐡 ∈ dom card)
47 dffn4 6809 . . . . . . . . . . . . . 14 (𝑓 Fn 𝐡 ↔ 𝑓:𝐡–ontoβ†’ran 𝑓)
483, 47sylib 217 . . . . . . . . . . . . 13 (𝑓:𝐡⟢𝐴 β†’ 𝑓:𝐡–ontoβ†’ran 𝑓)
49 fodomnum 10049 . . . . . . . . . . . . 13 (𝐡 ∈ dom card β†’ (𝑓:𝐡–ontoβ†’ran 𝑓 β†’ ran 𝑓 β‰Ό 𝐡))
5046, 48, 49syl2im 40 . . . . . . . . . . . 12 (𝐡 ∈ On β†’ (𝑓:𝐡⟢𝐴 β†’ ran 𝑓 β‰Ό 𝐡))
5150imp 408 . . . . . . . . . . 11 ((𝐡 ∈ On ∧ 𝑓:𝐡⟢𝐴) β†’ ran 𝑓 β‰Ό 𝐡)
52513adant1 1131 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐡 ∈ On ∧ 𝑓:𝐡⟢𝐴) β†’ ran 𝑓 β‰Ό 𝐡)
53 endomtr 9005 . . . . . . . . . 10 (((cardβ€˜ran 𝑓) β‰ˆ ran 𝑓 ∧ ran 𝑓 β‰Ό 𝐡) β†’ (cardβ€˜ran 𝑓) β‰Ό 𝐡)
5445, 52, 53syl2anc 585 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐡 ∈ On ∧ 𝑓:𝐡⟢𝐴) β†’ (cardβ€˜ran 𝑓) β‰Ό 𝐡)
55 cardon 9936 . . . . . . . . . . . 12 (cardβ€˜ran 𝑓) ∈ On
56 onenon 9941 . . . . . . . . . . . 12 ((cardβ€˜ran 𝑓) ∈ On β†’ (cardβ€˜ran 𝑓) ∈ dom card)
5755, 56ax-mp 5 . . . . . . . . . . 11 (cardβ€˜ran 𝑓) ∈ dom card
58 carddom2 9969 . . . . . . . . . . 11 (((cardβ€˜ran 𝑓) ∈ dom card ∧ 𝐡 ∈ dom card) β†’ ((cardβ€˜(cardβ€˜ran 𝑓)) βŠ† (cardβ€˜π΅) ↔ (cardβ€˜ran 𝑓) β‰Ό 𝐡))
5957, 46, 58sylancr 588 . . . . . . . . . 10 (𝐡 ∈ On β†’ ((cardβ€˜(cardβ€˜ran 𝑓)) βŠ† (cardβ€˜π΅) ↔ (cardβ€˜ran 𝑓) β‰Ό 𝐡))
60593ad2ant2 1135 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐡 ∈ On ∧ 𝑓:𝐡⟢𝐴) β†’ ((cardβ€˜(cardβ€˜ran 𝑓)) βŠ† (cardβ€˜π΅) ↔ (cardβ€˜ran 𝑓) β‰Ό 𝐡))
6154, 60mpbird 257 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐡 ∈ On ∧ 𝑓:𝐡⟢𝐴) β†’ (cardβ€˜(cardβ€˜ran 𝑓)) βŠ† (cardβ€˜π΅))
62 cardonle 9949 . . . . . . . . 9 (𝐡 ∈ On β†’ (cardβ€˜π΅) βŠ† 𝐡)
63623ad2ant2 1135 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐡 ∈ On ∧ 𝑓:𝐡⟢𝐴) β†’ (cardβ€˜π΅) βŠ† 𝐡)
6461, 63sstrd 3992 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐡 ∈ On ∧ 𝑓:𝐡⟢𝐴) β†’ (cardβ€˜(cardβ€˜ran 𝑓)) βŠ† 𝐡)
6538, 64eqsstrrid 4031 . . . . . 6 ((𝐴 ∈ On ∧ 𝐡 ∈ On ∧ 𝑓:𝐡⟢𝐴) β†’ (cardβ€˜ran 𝑓) βŠ† 𝐡)
66653expa 1119 . . . . 5 (((𝐴 ∈ On ∧ 𝐡 ∈ On) ∧ 𝑓:𝐡⟢𝐴) β†’ (cardβ€˜ran 𝑓) βŠ† 𝐡)
6766adantrr 716 . . . 4 (((𝐴 ∈ On ∧ 𝐡 ∈ On) ∧ (𝑓:𝐡⟢𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝐡 𝑧 βŠ† (π‘“β€˜π‘€))) β†’ (cardβ€˜ran 𝑓) βŠ† 𝐡)
6837, 67sstrd 3992 . . 3 (((𝐴 ∈ On ∧ 𝐡 ∈ On) ∧ (𝑓:𝐡⟢𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝐡 𝑧 βŠ† (π‘“β€˜π‘€))) β†’ (cfβ€˜π΄) βŠ† 𝐡)
6968ex 414 . 2 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ ((𝑓:𝐡⟢𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝐡 𝑧 βŠ† (π‘“β€˜π‘€)) β†’ (cfβ€˜π΄) βŠ† 𝐡))
7069exlimdv 1937 1 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (βˆƒπ‘“(𝑓:𝐡⟢𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝐡 𝑧 βŠ† (π‘“β€˜π‘€)) β†’ (cfβ€˜π΄) βŠ† 𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {cab 2710  βˆ€wral 3062  βˆƒwrex 3071  Vcvv 3475   βŠ† wss 3948  βˆ© cint 4950   class class class wbr 5148  dom cdm 5676  ran crn 5677  Oncon0 6362   Fn wfn 6536  βŸΆwf 6537  β€“ontoβ†’wfo 6539  β€˜cfv 6541   β‰ˆ cen 8933   β‰Ό cdom 8934  cardccrd 9927  cfccf 9929
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722
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 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-er 8700  df-map 8819  df-en 8937  df-dom 8938  df-sdom 8939  df-card 9931  df-cf 9933  df-acn 9934
This theorem is referenced by:  cfsmolem  10262  cfcoflem  10264  cfcof  10266  inar1  10767
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