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 42242
Description: Lemma for eldioph2 42243. 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 13964 . . . 4 (1...𝑁) ∈ Fin
3 difinf 9335 . . . 4 ((Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) ∈ Fin) β†’ Β¬ (𝑆 βˆ– (1...𝑁)) ∈ Fin)
41, 2, 3sylancl 584 . . 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 13964 . . . 4 (1...𝐴) ∈ Fin
6 diffi 9197 . . . 4 ((1...𝐴) ∈ Fin β†’ ((1...𝐴) βˆ– (1...𝑁)) ∈ Fin)
75, 6ax-mp 5 . . 3 ((1...𝐴) βˆ– (1...𝑁)) ∈ Fin
8 isinffi 10010 . . 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 584 . 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 6843 . . . . . . . 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 480 . . . . . . 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 6870 . . . . . . . 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 4469 . . . . . . . 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 6787 . . . . . . . . . . . 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 6725 . . . . . . . . . . 11 (π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁)) β†’ ran π‘Ž βŠ† (𝑆 βˆ– (1...𝑁)))
1817adantl 480 . . . . . . . . . 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 4231 . . . . . . . . 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 4469 . . . . . . . . 9 ((𝑆 βˆ– (1...𝑁)) ∩ (1...𝑁)) = βˆ…
2119, 20sseqtrdi 4024 . . . . . . . 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 4395 . . . . . . . 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 6851 . . . . . . 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 6831 . . . . . 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 4147 . . . . . . 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 13568 . . . . . . . . 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 4478 . . . . . . . 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 2777 . . . . . 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 6783 . . . . . 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 4128 . . . . . 6 (π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁)) β†’ ran π‘Ž βŠ† 𝑆)
3938adantl 480 . . . . 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 4181 . . . 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 6792 . . . 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 582 . . 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 5995 . . . 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 6012 . . . . . . . 8 dom (π‘Ž β†Ύ (1...𝑁)) = ((1...𝑁) ∩ dom π‘Ž)
46 incom 4196 . . . . . . . . 9 ((1...𝑁) ∩ dom π‘Ž) = (dom π‘Ž ∩ (1...𝑁))
47 f1dm 6791 . . . . . . . . . . . 12 (π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁)) β†’ dom π‘Ž = ((1...𝐴) βˆ– (1...𝑁)))
4847adantl 480 . . . . . . . . . . 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 4206 . . . . . . . . . 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 2781 . . . . . . . . 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 2777 . . . . . . . 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 2777 . . . . . . 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 6006 . . . . . . . 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 6010 . . . . . . 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 4158 . . . . 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 4147 . . . . . 6 (βˆ… βˆͺ ( I β†Ύ (1...𝑁))) = (( I β†Ύ (1...𝑁)) βˆͺ βˆ…)
61 un0 4387 . . . . . 6 (( I β†Ύ (1...𝑁)) βˆͺ βˆ…) = ( I β†Ύ (1...𝑁))
6260, 61eqtri 2753 . . . . 5 (βˆ… βˆͺ ( I β†Ύ (1...𝑁))) = ( I β†Ύ (1...𝑁))
6359, 62eqtrdi 2781 . . . 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 2777 . . 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 3467 . . . . 5 π‘Ž ∈ V
66 ovex 7446 . . . . . 6 (1...𝑁) ∈ V
67 resiexg 7914 . . . . . 6 ((1...𝑁) ∈ V β†’ ( I β†Ύ (1...𝑁)) ∈ V)
6866, 67ax-mp 5 . . . . 5 ( I β†Ύ (1...𝑁)) ∈ V
6965, 68unex 7743 . . . 4 (π‘Ž βˆͺ ( I β†Ύ (1...𝑁))) ∈ V
70 f1eq1 6782 . . . . 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 5974 . . . . . 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 2727 . . . . 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 630 . . . 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 3587 . . 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 582 . 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 1930 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 394   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098  Vcvv 3463   βˆ– cdif 3938   βˆͺ cun 3939   ∩ cin 3940   βŠ† wss 3941  βˆ…c0 4319   I cid 5570  dom cdm 5673  ran crn 5674   β†Ύ cres 5675  Rel wrel 5678  β€“1-1β†’wf1 6540  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  (class class class)co 7413  Fincfn 8957  1c1 11134  β„•0cn0 12497  β„€β‰₯cuz 12847  ...cfz 13511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-card 9957  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-nn 12238  df-n0 12498  df-z 12584  df-uz 12848  df-fz 13512
This theorem is referenced by:  eldioph2b  42244
  Copyright terms: Public domain W3C validator