Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eldioph2lem2 Structured version   Visualization version   GIF version
Theorem eldioph2lem2 41484
Description: Lemma for eldioph2 41485. Construct necessary renaming function for one direction. (Contributed by Stefan O'Rear, 8-Oct-2014.)
Assertion
Ref Expression
eldioph2lem2 (((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) β†’ βˆƒπ‘(𝑐:(1...𝐴)–1-1→𝑆 ∧ (𝑐 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁))))
Distinct variable groups:   𝑁,𝑐   𝑆,𝑐   𝐴,𝑐
Proof of Theorem eldioph2lem2
Dummy variable π‘Ž is distinct from all other variables.
StepHypRef Expression
1 simplr 767 . . . 4 (((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) β†’ Β¬ 𝑆 ∈ Fin)
2 fzfi 13933 . . . 4 (1...𝑁) ∈ Fin
3 difinf 9312 . . . 4 ((Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) ∈ Fin) β†’ Β¬ (𝑆 βˆ– (1...𝑁)) ∈ Fin)
41, 2, 3sylancl 586 . . 3 (((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) β†’ Β¬ (𝑆 βˆ– (1...𝑁)) ∈ Fin)
5 fzfi 13933 . . . 4 (1...𝐴) ∈ Fin
6 diffi 9175 . . . 4 ((1...𝐴) ∈ Fin β†’ ((1...𝐴) βˆ– (1...𝑁)) ∈ Fin)
75, 6ax-mp 5 . . 3 ((1...𝐴) βˆ– (1...𝑁)) ∈ Fin
8 isinffi 9983 . . 3 ((Β¬ (𝑆 βˆ– (1...𝑁)) ∈ Fin ∧ ((1...𝐴) βˆ– (1...𝑁)) ∈ Fin) β†’ βˆƒπ‘Ž π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁)))
94, 7, 8sylancl 586 . 2 (((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) β†’ βˆƒπ‘Ž π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁)))
10 f1f1orn 6841 . . . . . . . 8 (π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁)) β†’ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1-ontoβ†’ran π‘Ž)
1110adantl 482 . . . . . . 7 ((((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) ∧ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁))) β†’ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1-ontoβ†’ran π‘Ž)
12 f1oi 6868 . . . . . . . 8 ( I β†Ύ (1...𝑁)):(1...𝑁)–1-1-ontoβ†’(1...𝑁)
1312a1i 11 . . . . . . 7 ((((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) ∧ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁))) β†’ ( I β†Ύ (1...𝑁)):(1...𝑁)–1-1-ontoβ†’(1...𝑁))
14 disjdifr 4471 . . . . . . . 8 (((1...𝐴) βˆ– (1...𝑁)) ∩ (1...𝑁)) = βˆ…
1514a1i 11 . . . . . . 7 ((((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) ∧ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁))) β†’ (((1...𝐴) βˆ– (1...𝑁)) ∩ (1...𝑁)) = βˆ…)
16 f1f 6784 . . . . . . . . . . . 12 (π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁)) β†’ π‘Ž:((1...𝐴) βˆ– (1...𝑁))⟢(𝑆 βˆ– (1...𝑁)))
1716frnd 6722 . . . . . . . . . . 11 (π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁)) β†’ ran π‘Ž βŠ† (𝑆 βˆ– (1...𝑁)))
1817adantl 482 . . . . . . . . . 10 ((((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) ∧ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁))) β†’ ran π‘Ž βŠ† (𝑆 βˆ– (1...𝑁)))
1918ssrind 4234 . . . . . . . . 9 ((((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) ∧ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁))) β†’ (ran π‘Ž ∩ (1...𝑁)) βŠ† ((𝑆 βˆ– (1...𝑁)) ∩ (1...𝑁)))
20 disjdifr 4471 . . . . . . . . 9 ((𝑆 βˆ– (1...𝑁)) ∩ (1...𝑁)) = βˆ…
2119, 20sseqtrdi 4031 . . . . . . . 8 ((((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) ∧ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁))) β†’ (ran π‘Ž ∩ (1...𝑁)) βŠ† βˆ…)
22 ss0 4397 . . . . . . . 8 ((ran π‘Ž ∩ (1...𝑁)) βŠ† βˆ… β†’ (ran π‘Ž ∩ (1...𝑁)) = βˆ…)
2321, 22syl 17 . . . . . . 7 ((((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) ∧ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁))) β†’ (ran π‘Ž ∩ (1...𝑁)) = βˆ…)
24 f1oun 6849 . . . . . . 7 (((π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1-ontoβ†’ran π‘Ž ∧ ( I β†Ύ (1...𝑁)):(1...𝑁)–1-1-ontoβ†’(1...𝑁)) ∧ ((((1...𝐴) βˆ– (1...𝑁)) ∩ (1...𝑁)) = βˆ… ∧ (ran π‘Ž ∩ (1...𝑁)) = βˆ…)) β†’ (π‘Ž βˆͺ ( I β†Ύ (1...𝑁))):(((1...𝐴) βˆ– (1...𝑁)) βˆͺ (1...𝑁))–1-1-ontoβ†’(ran π‘Ž βˆͺ (1...𝑁)))
2511, 13, 15, 23, 24syl22anc 837 . . . . . 6 ((((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) ∧ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁))) β†’ (π‘Ž βˆͺ ( I β†Ύ (1...𝑁))):(((1...𝐴) βˆ– (1...𝑁)) βˆͺ (1...𝑁))–1-1-ontoβ†’(ran π‘Ž βˆͺ (1...𝑁)))
26 f1of1 6829 . . . . . 6 ((π‘Ž βˆͺ ( I β†Ύ (1...𝑁))):(((1...𝐴) βˆ– (1...𝑁)) βˆͺ (1...𝑁))–1-1-ontoβ†’(ran π‘Ž βˆͺ (1...𝑁)) β†’ (π‘Ž βˆͺ ( I β†Ύ (1...𝑁))):(((1...𝐴) βˆ– (1...𝑁)) βˆͺ (1...𝑁))–1-1β†’(ran π‘Ž βˆͺ (1...𝑁)))
2725, 26syl 17 . . . . 5 ((((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) ∧ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁))) β†’ (π‘Ž βˆͺ ( I β†Ύ (1...𝑁))):(((1...𝐴) βˆ– (1...𝑁)) βˆͺ (1...𝑁))–1-1β†’(ran π‘Ž βˆͺ (1...𝑁)))
28 uncom 4152 . . . . . . 7 (((1...𝐴) βˆ– (1...𝑁)) βˆͺ (1...𝑁)) = ((1...𝑁) βˆͺ ((1...𝐴) βˆ– (1...𝑁)))
29 simplrr 776 . . . . . . . . 9 ((((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) ∧ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁))) β†’ 𝐴 ∈ (β„€β‰₯β€˜π‘))
30 fzss2 13537 . . . . . . . . 9 (𝐴 ∈ (β„€β‰₯β€˜π‘) β†’ (1...𝑁) βŠ† (1...𝐴))
3129, 30syl 17 . . . . . . . 8 ((((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) ∧ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁))) β†’ (1...𝑁) βŠ† (1...𝐴))
32 undif 4480 . . . . . . . 8 ((1...𝑁) βŠ† (1...𝐴) ↔ ((1...𝑁) βˆͺ ((1...𝐴) βˆ– (1...𝑁))) = (1...𝐴))
3331, 32sylib 217 . . . . . . 7 ((((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) ∧ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁))) β†’ ((1...𝑁) βˆͺ ((1...𝐴) βˆ– (1...𝑁))) = (1...𝐴))
3428, 33eqtrid 2784 . . . . . 6 ((((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) ∧ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁))) β†’ (((1...𝐴) βˆ– (1...𝑁)) βˆͺ (1...𝑁)) = (1...𝐴))
35 f1eq2 6780 . . . . . 6 ((((1...𝐴) βˆ– (1...𝑁)) βˆͺ (1...𝑁)) = (1...𝐴) β†’ ((π‘Ž βˆͺ ( I β†Ύ (1...𝑁))):(((1...𝐴) βˆ– (1...𝑁)) βˆͺ (1...𝑁))–1-1β†’(ran π‘Ž βˆͺ (1...𝑁)) ↔ (π‘Ž βˆͺ ( I β†Ύ (1...𝑁))):(1...𝐴)–1-1β†’(ran π‘Ž βˆͺ (1...𝑁))))
3634, 35syl 17 . . . . 5 ((((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) ∧ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁))) β†’ ((π‘Ž βˆͺ ( I β†Ύ (1...𝑁))):(((1...𝐴) βˆ– (1...𝑁)) βˆͺ (1...𝑁))–1-1β†’(ran π‘Ž βˆͺ (1...𝑁)) ↔ (π‘Ž βˆͺ ( I β†Ύ (1...𝑁))):(1...𝐴)–1-1β†’(ran π‘Ž βˆͺ (1...𝑁))))
3727, 36mpbid 231 . . . 4 ((((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) ∧ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁))) β†’ (π‘Ž βˆͺ ( I β†Ύ (1...𝑁))):(1...𝐴)–1-1β†’(ran π‘Ž βˆͺ (1...𝑁)))
3817difss2d 4133 . . . . . 6 (π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁)) β†’ ran π‘Ž βŠ† 𝑆)
3938adantl 482 . . . . 5 ((((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) ∧ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁))) β†’ ran π‘Ž βŠ† 𝑆)
40 simplrl 775 . . . . 5 ((((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) ∧ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁))) β†’ (1...𝑁) βŠ† 𝑆)
4139, 40unssd 4185 . . . 4 ((((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) ∧ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁))) β†’ (ran π‘Ž βˆͺ (1...𝑁)) βŠ† 𝑆)
42 f1ss 6790 . . . 4 (((π‘Ž βˆͺ ( I β†Ύ (1...𝑁))):(1...𝐴)–1-1β†’(ran π‘Ž βˆͺ (1...𝑁)) ∧ (ran π‘Ž βˆͺ (1...𝑁)) βŠ† 𝑆) β†’ (π‘Ž βˆͺ ( I β†Ύ (1...𝑁))):(1...𝐴)–1-1→𝑆)
4337, 41, 42syl2anc 584 . . 3 ((((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) ∧ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁))) β†’ (π‘Ž βˆͺ ( I β†Ύ (1...𝑁))):(1...𝐴)–1-1→𝑆)
44 resundir 5994 . . . 4 ((π‘Ž βˆͺ ( I β†Ύ (1...𝑁))) β†Ύ (1...𝑁)) = ((π‘Ž β†Ύ (1...𝑁)) βˆͺ (( I β†Ύ (1...𝑁)) β†Ύ (1...𝑁)))
45 dmres 6001 . . . . . . . 8 dom (π‘Ž β†Ύ (1...𝑁)) = ((1...𝑁) ∩ dom π‘Ž)
46 incom 4200 . . . . . . . . 9 ((1...𝑁) ∩ dom π‘Ž) = (dom π‘Ž ∩ (1...𝑁))
47 f1dm 6788 . . . . . . . . . . . 12 (π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁)) β†’ dom π‘Ž = ((1...𝐴) βˆ– (1...𝑁)))
4847adantl 482 . . . . . . . . . . 11 ((((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) ∧ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁))) β†’ dom π‘Ž = ((1...𝐴) βˆ– (1...𝑁)))
4948ineq1d 4210 . . . . . . . . . 10 ((((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) ∧ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁))) β†’ (dom π‘Ž ∩ (1...𝑁)) = (((1...𝐴) βˆ– (1...𝑁)) ∩ (1...𝑁)))
5049, 14eqtrdi 2788 . . . . . . . . 9 ((((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) ∧ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁))) β†’ (dom π‘Ž ∩ (1...𝑁)) = βˆ…)
5146, 50eqtrid 2784 . . . . . . . 8 ((((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) ∧ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁))) β†’ ((1...𝑁) ∩ dom π‘Ž) = βˆ…)
5245, 51eqtrid 2784 . . . . . . 7 ((((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) ∧ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁))) β†’ dom (π‘Ž β†Ύ (1...𝑁)) = βˆ…)
53 relres 6008 . . . . . . . 8 Rel (π‘Ž β†Ύ (1...𝑁))
54 reldm0 5925 . . . . . . . 8 (Rel (π‘Ž β†Ύ (1...𝑁)) β†’ ((π‘Ž β†Ύ (1...𝑁)) = βˆ… ↔ dom (π‘Ž β†Ύ (1...𝑁)) = βˆ…))
5553, 54ax-mp 5 . . . . . . 7 ((π‘Ž β†Ύ (1...𝑁)) = βˆ… ↔ dom (π‘Ž β†Ύ (1...𝑁)) = βˆ…)
5652, 55sylibr 233 . . . . . 6 ((((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) ∧ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁))) β†’ (π‘Ž β†Ύ (1...𝑁)) = βˆ…)
57 residm 6012 . . . . . . 7 (( I β†Ύ (1...𝑁)) β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁))
5857a1i 11 . . . . . 6 ((((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) ∧ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁))) β†’ (( I β†Ύ (1...𝑁)) β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))
5956, 58uneq12d 4163 . . . . 5 ((((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) ∧ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁))) β†’ ((π‘Ž β†Ύ (1...𝑁)) βˆͺ (( I β†Ύ (1...𝑁)) β†Ύ (1...𝑁))) = (βˆ… βˆͺ ( I β†Ύ (1...𝑁))))
60 uncom 4152 . . . . . 6 (βˆ… βˆͺ ( I β†Ύ (1...𝑁))) = (( I β†Ύ (1...𝑁)) βˆͺ βˆ…)
61 un0 4389 . . . . . 6 (( I β†Ύ (1...𝑁)) βˆͺ βˆ…) = ( I β†Ύ (1...𝑁))
6260, 61eqtri 2760 . . . . 5 (βˆ… βˆͺ ( I β†Ύ (1...𝑁))) = ( I β†Ύ (1...𝑁))
6359, 62eqtrdi 2788 . . . 4 ((((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) ∧ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁))) β†’ ((π‘Ž β†Ύ (1...𝑁)) βˆͺ (( I β†Ύ (1...𝑁)) β†Ύ (1...𝑁))) = ( I β†Ύ (1...𝑁)))
6444, 63eqtrid 2784 . . 3 ((((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) ∧ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁))) β†’ ((π‘Ž βˆͺ ( I β†Ύ (1...𝑁))) β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))
65 vex 3478 . . . . 5 π‘Ž ∈ V
66 ovex 7438 . . . . . 6 (1...𝑁) ∈ V
67 resiexg 7901 . . . . . 6 ((1...𝑁) ∈ V β†’ ( I β†Ύ (1...𝑁)) ∈ V)
6866, 67ax-mp 5 . . . . 5 ( I β†Ύ (1...𝑁)) ∈ V
6965, 68unex 7729 . . . 4 (π‘Ž βˆͺ ( I β†Ύ (1...𝑁))) ∈ V
70 f1eq1 6779 . . . . 5 (𝑐 = (π‘Ž βˆͺ ( I β†Ύ (1...𝑁))) β†’ (𝑐:(1...𝐴)–1-1→𝑆 ↔ (π‘Ž βˆͺ ( I β†Ύ (1...𝑁))):(1...𝐴)–1-1→𝑆))
71 reseq1 5973 . . . . . 6 (𝑐 = (π‘Ž βˆͺ ( I β†Ύ (1...𝑁))) β†’ (𝑐 β†Ύ (1...𝑁)) = ((π‘Ž βˆͺ ( I β†Ύ (1...𝑁))) β†Ύ (1...𝑁)))
7271eqeq1d 2734 . . . . 5 (𝑐 = (π‘Ž βˆͺ ( I β†Ύ (1...𝑁))) β†’ ((𝑐 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)) ↔ ((π‘Ž βˆͺ ( I β†Ύ (1...𝑁))) β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁))))
7370, 72anbi12d 631 . . . 4 (𝑐 = (π‘Ž βˆͺ ( I β†Ύ (1...𝑁))) β†’ ((𝑐:(1...𝐴)–1-1→𝑆 ∧ (𝑐 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁))) ↔ ((π‘Ž βˆͺ ( I β†Ύ (1...𝑁))):(1...𝐴)–1-1→𝑆 ∧ ((π‘Ž βˆͺ ( I β†Ύ (1...𝑁))) β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁)))))
7469, 73spcev 3596 . . 3 (((π‘Ž βˆͺ ( I β†Ύ (1...𝑁))):(1...𝐴)–1-1→𝑆 ∧ ((π‘Ž βˆͺ ( I β†Ύ (1...𝑁))) β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁))) β†’ βˆƒπ‘(𝑐:(1...𝐴)–1-1→𝑆 ∧ (𝑐 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁))))
7543, 64, 74syl2anc 584 . 2 ((((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) ∧ π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁))) β†’ βˆƒπ‘(𝑐:(1...𝐴)–1-1→𝑆 ∧ (𝑐 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁))))
769, 75exlimddv 1938 1 (((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) β†’ βˆƒπ‘(𝑐:(1...𝐴)–1-1→𝑆 ∧ (𝑐 β†Ύ (1...𝑁)) = ( I β†Ύ (1...𝑁))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  Vcvv 3474   βˆ– cdif 3944   βˆͺ cun 3945   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4321   I cid 5572  dom cdm 5675  ran crn 5676   β†Ύ cres 5677  Rel wrel 5680  β€“1-1β†’wf1 6537  β€“1-1-ontoβ†’wf1o 6539  β€˜cfv 6540  (class class class)co 7405  Fincfn 8935  1c1 11107  β„•0cn0 12468  β„€β‰₯cuz 12818  ...cfz 13480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  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-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 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-n0 12469  df-z 12555  df-uz 12819  df-fz 13481
This theorem is referenced by:  eldioph2b  41486
  Copyright terms: Public domain W3C validator