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Theorem indexfi 9360
Description: If for every element of a finite indexing set 𝐴 there exists a corresponding element of another set 𝐡, then there exists a finite subset of 𝐡 consisting only of those elements which are indexed by 𝐴. Proven without the Axiom of Choice, unlike indexdom 36650. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
Assertion
Ref Expression
indexfi ((𝐴 ∈ Fin ∧ 𝐡 ∈ 𝑀 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 πœ‘) β†’ βˆƒπ‘ ∈ Fin (𝑐 βŠ† 𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝑐 πœ‘ ∧ βˆ€π‘¦ ∈ 𝑐 βˆƒπ‘₯ ∈ 𝐴 πœ‘))
Distinct variable groups:   π‘₯,𝑐,𝑦,𝐴   𝐡,𝑐,π‘₯,𝑦   πœ‘,𝑐
Allowed substitution hints:   πœ‘(π‘₯,𝑦)   𝑀(π‘₯,𝑦,𝑐)

Proof of Theorem indexfi
Dummy variables 𝑓 𝑀 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1918 . . . . . 6 β„²π‘§πœ‘
2 nfsbc1v 3798 . . . . . 6 Ⅎ𝑦[𝑧 / 𝑦]πœ‘
3 sbceq1a 3789 . . . . . 6 (𝑦 = 𝑧 β†’ (πœ‘ ↔ [𝑧 / 𝑦]πœ‘))
41, 2, 3cbvrexw 3305 . . . . 5 (βˆƒπ‘¦ ∈ 𝐡 πœ‘ ↔ βˆƒπ‘§ ∈ 𝐡 [𝑧 / 𝑦]πœ‘)
54ralbii 3094 . . . 4 (βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 πœ‘ ↔ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘§ ∈ 𝐡 [𝑧 / 𝑦]πœ‘)
6 dfsbcq 3780 . . . . 5 (𝑧 = (π‘“β€˜π‘₯) β†’ ([𝑧 / 𝑦]πœ‘ ↔ [(π‘“β€˜π‘₯) / 𝑦]πœ‘))
76ac6sfi 9287 . . . 4 ((𝐴 ∈ Fin ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘§ ∈ 𝐡 [𝑧 / 𝑦]πœ‘) β†’ βˆƒπ‘“(𝑓:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 [(π‘“β€˜π‘₯) / 𝑦]πœ‘))
85, 7sylan2b 595 . . 3 ((𝐴 ∈ Fin ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 πœ‘) β†’ βˆƒπ‘“(𝑓:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 [(π‘“β€˜π‘₯) / 𝑦]πœ‘))
9 simpll 766 . . . . 5 (((𝐴 ∈ Fin ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 πœ‘) ∧ (𝑓:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 [(π‘“β€˜π‘₯) / 𝑦]πœ‘)) β†’ 𝐴 ∈ Fin)
10 ffn 6718 . . . . . . 7 (𝑓:𝐴⟢𝐡 β†’ 𝑓 Fn 𝐴)
1110ad2antrl 727 . . . . . 6 (((𝐴 ∈ Fin ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 πœ‘) ∧ (𝑓:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 [(π‘“β€˜π‘₯) / 𝑦]πœ‘)) β†’ 𝑓 Fn 𝐴)
12 dffn4 6812 . . . . . 6 (𝑓 Fn 𝐴 ↔ 𝑓:𝐴–ontoβ†’ran 𝑓)
1311, 12sylib 217 . . . . 5 (((𝐴 ∈ Fin ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 πœ‘) ∧ (𝑓:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 [(π‘“β€˜π‘₯) / 𝑦]πœ‘)) β†’ 𝑓:𝐴–ontoβ†’ran 𝑓)
14 fofi 9338 . . . . 5 ((𝐴 ∈ Fin ∧ 𝑓:𝐴–ontoβ†’ran 𝑓) β†’ ran 𝑓 ∈ Fin)
159, 13, 14syl2anc 585 . . . 4 (((𝐴 ∈ Fin ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 πœ‘) ∧ (𝑓:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 [(π‘“β€˜π‘₯) / 𝑦]πœ‘)) β†’ ran 𝑓 ∈ Fin)
16 frn 6725 . . . . 5 (𝑓:𝐴⟢𝐡 β†’ ran 𝑓 βŠ† 𝐡)
1716ad2antrl 727 . . . 4 (((𝐴 ∈ Fin ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 πœ‘) ∧ (𝑓:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 [(π‘“β€˜π‘₯) / 𝑦]πœ‘)) β†’ ran 𝑓 βŠ† 𝐡)
18 fnfvelrn 7083 . . . . . . . . 9 ((𝑓 Fn 𝐴 ∧ π‘₯ ∈ 𝐴) β†’ (π‘“β€˜π‘₯) ∈ ran 𝑓)
1910, 18sylan 581 . . . . . . . 8 ((𝑓:𝐴⟢𝐡 ∧ π‘₯ ∈ 𝐴) β†’ (π‘“β€˜π‘₯) ∈ ran 𝑓)
20 rspesbca 3876 . . . . . . . . 9 (((π‘“β€˜π‘₯) ∈ ran 𝑓 ∧ [(π‘“β€˜π‘₯) / 𝑦]πœ‘) β†’ βˆƒπ‘¦ ∈ ran π‘“πœ‘)
2120ex 414 . . . . . . . 8 ((π‘“β€˜π‘₯) ∈ ran 𝑓 β†’ ([(π‘“β€˜π‘₯) / 𝑦]πœ‘ β†’ βˆƒπ‘¦ ∈ ran π‘“πœ‘))
2219, 21syl 17 . . . . . . 7 ((𝑓:𝐴⟢𝐡 ∧ π‘₯ ∈ 𝐴) β†’ ([(π‘“β€˜π‘₯) / 𝑦]πœ‘ β†’ βˆƒπ‘¦ ∈ ran π‘“πœ‘))
2322ralimdva 3168 . . . . . 6 (𝑓:𝐴⟢𝐡 β†’ (βˆ€π‘₯ ∈ 𝐴 [(π‘“β€˜π‘₯) / 𝑦]πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ ran π‘“πœ‘))
2423imp 408 . . . . 5 ((𝑓:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 [(π‘“β€˜π‘₯) / 𝑦]πœ‘) β†’ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ ran π‘“πœ‘)
2524adantl 483 . . . 4 (((𝐴 ∈ Fin ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 πœ‘) ∧ (𝑓:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 [(π‘“β€˜π‘₯) / 𝑦]πœ‘)) β†’ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ ran π‘“πœ‘)
26 simpr 486 . . . . . . . 8 ((((𝐴 ∈ Fin ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 πœ‘) ∧ (𝑓:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 [(π‘“β€˜π‘₯) / 𝑦]πœ‘)) ∧ 𝑀 ∈ 𝐴) β†’ 𝑀 ∈ 𝐴)
27 simprr 772 . . . . . . . . . 10 (((𝐴 ∈ Fin ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 πœ‘) ∧ (𝑓:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 [(π‘“β€˜π‘₯) / 𝑦]πœ‘)) β†’ βˆ€π‘₯ ∈ 𝐴 [(π‘“β€˜π‘₯) / 𝑦]πœ‘)
28 nfv 1918 . . . . . . . . . . 11 Ⅎ𝑀[(π‘“β€˜π‘₯) / 𝑦]πœ‘
29 nfsbc1v 3798 . . . . . . . . . . 11 β„²π‘₯[𝑀 / π‘₯][(π‘“β€˜π‘€) / 𝑦]πœ‘
30 fveq2 6892 . . . . . . . . . . . . 13 (π‘₯ = 𝑀 β†’ (π‘“β€˜π‘₯) = (π‘“β€˜π‘€))
3130sbceq1d 3783 . . . . . . . . . . . 12 (π‘₯ = 𝑀 β†’ ([(π‘“β€˜π‘₯) / 𝑦]πœ‘ ↔ [(π‘“β€˜π‘€) / 𝑦]πœ‘))
32 sbceq1a 3789 . . . . . . . . . . . 12 (π‘₯ = 𝑀 β†’ ([(π‘“β€˜π‘€) / 𝑦]πœ‘ ↔ [𝑀 / π‘₯][(π‘“β€˜π‘€) / 𝑦]πœ‘))
3331, 32bitrd 279 . . . . . . . . . . 11 (π‘₯ = 𝑀 β†’ ([(π‘“β€˜π‘₯) / 𝑦]πœ‘ ↔ [𝑀 / π‘₯][(π‘“β€˜π‘€) / 𝑦]πœ‘))
3428, 29, 33cbvralw 3304 . . . . . . . . . 10 (βˆ€π‘₯ ∈ 𝐴 [(π‘“β€˜π‘₯) / 𝑦]πœ‘ ↔ βˆ€π‘€ ∈ 𝐴 [𝑀 / π‘₯][(π‘“β€˜π‘€) / 𝑦]πœ‘)
3527, 34sylib 217 . . . . . . . . 9 (((𝐴 ∈ Fin ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 πœ‘) ∧ (𝑓:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 [(π‘“β€˜π‘₯) / 𝑦]πœ‘)) β†’ βˆ€π‘€ ∈ 𝐴 [𝑀 / π‘₯][(π‘“β€˜π‘€) / 𝑦]πœ‘)
3635r19.21bi 3249 . . . . . . . 8 ((((𝐴 ∈ Fin ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 πœ‘) ∧ (𝑓:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 [(π‘“β€˜π‘₯) / 𝑦]πœ‘)) ∧ 𝑀 ∈ 𝐴) β†’ [𝑀 / π‘₯][(π‘“β€˜π‘€) / 𝑦]πœ‘)
37 rspesbca 3876 . . . . . . . 8 ((𝑀 ∈ 𝐴 ∧ [𝑀 / π‘₯][(π‘“β€˜π‘€) / 𝑦]πœ‘) β†’ βˆƒπ‘₯ ∈ 𝐴 [(π‘“β€˜π‘€) / 𝑦]πœ‘)
3826, 36, 37syl2anc 585 . . . . . . 7 ((((𝐴 ∈ Fin ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 πœ‘) ∧ (𝑓:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 [(π‘“β€˜π‘₯) / 𝑦]πœ‘)) ∧ 𝑀 ∈ 𝐴) β†’ βˆƒπ‘₯ ∈ 𝐴 [(π‘“β€˜π‘€) / 𝑦]πœ‘)
3938ralrimiva 3147 . . . . . 6 (((𝐴 ∈ Fin ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 πœ‘) ∧ (𝑓:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 [(π‘“β€˜π‘₯) / 𝑦]πœ‘)) β†’ βˆ€π‘€ ∈ 𝐴 βˆƒπ‘₯ ∈ 𝐴 [(π‘“β€˜π‘€) / 𝑦]πœ‘)
40 dfsbcq 3780 . . . . . . . . 9 (𝑧 = (π‘“β€˜π‘€) β†’ ([𝑧 / 𝑦]πœ‘ ↔ [(π‘“β€˜π‘€) / 𝑦]πœ‘))
4140rexbidv 3179 . . . . . . . 8 (𝑧 = (π‘“β€˜π‘€) β†’ (βˆƒπ‘₯ ∈ 𝐴 [𝑧 / 𝑦]πœ‘ ↔ βˆƒπ‘₯ ∈ 𝐴 [(π‘“β€˜π‘€) / 𝑦]πœ‘))
4241ralrn 7090 . . . . . . 7 (𝑓 Fn 𝐴 β†’ (βˆ€π‘§ ∈ ran π‘“βˆƒπ‘₯ ∈ 𝐴 [𝑧 / 𝑦]πœ‘ ↔ βˆ€π‘€ ∈ 𝐴 βˆƒπ‘₯ ∈ 𝐴 [(π‘“β€˜π‘€) / 𝑦]πœ‘))
4311, 42syl 17 . . . . . 6 (((𝐴 ∈ Fin ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 πœ‘) ∧ (𝑓:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 [(π‘“β€˜π‘₯) / 𝑦]πœ‘)) β†’ (βˆ€π‘§ ∈ ran π‘“βˆƒπ‘₯ ∈ 𝐴 [𝑧 / 𝑦]πœ‘ ↔ βˆ€π‘€ ∈ 𝐴 βˆƒπ‘₯ ∈ 𝐴 [(π‘“β€˜π‘€) / 𝑦]πœ‘))
4439, 43mpbird 257 . . . . 5 (((𝐴 ∈ Fin ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 πœ‘) ∧ (𝑓:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 [(π‘“β€˜π‘₯) / 𝑦]πœ‘)) β†’ βˆ€π‘§ ∈ ran π‘“βˆƒπ‘₯ ∈ 𝐴 [𝑧 / 𝑦]πœ‘)
45 nfv 1918 . . . . . 6 β„²π‘§βˆƒπ‘₯ ∈ 𝐴 πœ‘
46 nfcv 2904 . . . . . . 7 Ⅎ𝑦𝐴
4746, 2nfrexw 3311 . . . . . 6 β„²π‘¦βˆƒπ‘₯ ∈ 𝐴 [𝑧 / 𝑦]πœ‘
483rexbidv 3179 . . . . . 6 (𝑦 = 𝑧 β†’ (βˆƒπ‘₯ ∈ 𝐴 πœ‘ ↔ βˆƒπ‘₯ ∈ 𝐴 [𝑧 / 𝑦]πœ‘))
4945, 47, 48cbvralw 3304 . . . . 5 (βˆ€π‘¦ ∈ ran π‘“βˆƒπ‘₯ ∈ 𝐴 πœ‘ ↔ βˆ€π‘§ ∈ ran π‘“βˆƒπ‘₯ ∈ 𝐴 [𝑧 / 𝑦]πœ‘)
5044, 49sylibr 233 . . . 4 (((𝐴 ∈ Fin ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 πœ‘) ∧ (𝑓:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 [(π‘“β€˜π‘₯) / 𝑦]πœ‘)) β†’ βˆ€π‘¦ ∈ ran π‘“βˆƒπ‘₯ ∈ 𝐴 πœ‘)
51 sseq1 4008 . . . . . 6 (𝑐 = ran 𝑓 β†’ (𝑐 βŠ† 𝐡 ↔ ran 𝑓 βŠ† 𝐡))
52 rexeq 3322 . . . . . . 7 (𝑐 = ran 𝑓 β†’ (βˆƒπ‘¦ ∈ 𝑐 πœ‘ ↔ βˆƒπ‘¦ ∈ ran π‘“πœ‘))
5352ralbidv 3178 . . . . . 6 (𝑐 = ran 𝑓 β†’ (βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝑐 πœ‘ ↔ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ ran π‘“πœ‘))
54 raleq 3323 . . . . . 6 (𝑐 = ran 𝑓 β†’ (βˆ€π‘¦ ∈ 𝑐 βˆƒπ‘₯ ∈ 𝐴 πœ‘ ↔ βˆ€π‘¦ ∈ ran π‘“βˆƒπ‘₯ ∈ 𝐴 πœ‘))
5551, 53, 543anbi123d 1437 . . . . 5 (𝑐 = ran 𝑓 β†’ ((𝑐 βŠ† 𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝑐 πœ‘ ∧ βˆ€π‘¦ ∈ 𝑐 βˆƒπ‘₯ ∈ 𝐴 πœ‘) ↔ (ran 𝑓 βŠ† 𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ ran π‘“πœ‘ ∧ βˆ€π‘¦ ∈ ran π‘“βˆƒπ‘₯ ∈ 𝐴 πœ‘)))
5655rspcev 3613 . . . 4 ((ran 𝑓 ∈ Fin ∧ (ran 𝑓 βŠ† 𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ ran π‘“πœ‘ ∧ βˆ€π‘¦ ∈ ran π‘“βˆƒπ‘₯ ∈ 𝐴 πœ‘)) β†’ βˆƒπ‘ ∈ Fin (𝑐 βŠ† 𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝑐 πœ‘ ∧ βˆ€π‘¦ ∈ 𝑐 βˆƒπ‘₯ ∈ 𝐴 πœ‘))
5715, 17, 25, 50, 56syl13anc 1373 . . 3 (((𝐴 ∈ Fin ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 πœ‘) ∧ (𝑓:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 [(π‘“β€˜π‘₯) / 𝑦]πœ‘)) β†’ βˆƒπ‘ ∈ Fin (𝑐 βŠ† 𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝑐 πœ‘ ∧ βˆ€π‘¦ ∈ 𝑐 βˆƒπ‘₯ ∈ 𝐴 πœ‘))
588, 57exlimddv 1939 . 2 ((𝐴 ∈ Fin ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 πœ‘) β†’ βˆƒπ‘ ∈ Fin (𝑐 βŠ† 𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝑐 πœ‘ ∧ βˆ€π‘¦ ∈ 𝑐 βˆƒπ‘₯ ∈ 𝐴 πœ‘))
59583adant2 1132 1 ((𝐴 ∈ Fin ∧ 𝐡 ∈ 𝑀 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 πœ‘) β†’ βˆƒπ‘ ∈ Fin (𝑐 βŠ† 𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝑐 πœ‘ ∧ βˆ€π‘¦ ∈ 𝑐 βˆƒπ‘₯ ∈ 𝐴 πœ‘))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071  [wsbc 3778   βŠ† wss 3949  ran crn 5678   Fn wfn 6539  βŸΆwf 6540  β€“ontoβ†’wfo 6542  β€˜cfv 6544  Fincfn 8939
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-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-om 7856  df-1o 8466  df-er 8703  df-en 8940  df-dom 8941  df-fin 8943
This theorem is referenced by:  filbcmb  36656
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