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Theorem eldioph2lem2 42093
Description: Lemma for eldioph2 42094. 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 768 . . . 4 (((𝑁 ∈ β„•0 ∧ Β¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) βŠ† 𝑆 ∧ 𝐴 ∈ (β„€β‰₯β€˜π‘))) β†’ Β¬ 𝑆 ∈ Fin)
2 fzfi 13955 . . . 4 (1...𝑁) ∈ Fin
3 difinf 9330 . . . 4 ((Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) ∈ Fin) β†’ Β¬ (𝑆 βˆ– (1...𝑁)) ∈ Fin)
41, 2, 3sylancl 585 . . 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 13955 . . . 4 (1...𝐴) ∈ Fin
6 diffi 9193 . . . 4 ((1...𝐴) ∈ Fin β†’ ((1...𝐴) βˆ– (1...𝑁)) ∈ Fin)
75, 6ax-mp 5 . . 3 ((1...𝐴) βˆ– (1...𝑁)) ∈ Fin
8 isinffi 10001 . . 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 585 . 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 6844 . . . . . . . 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 481 . . . . . . 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 6871 . . . . . . . 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 4468 . . . . . . . 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 6724 . . . . . . . . . . 11 (π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁)) β†’ ran π‘Ž βŠ† (𝑆 βˆ– (1...𝑁)))
1817adantl 481 . . . . . . . . . 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 4468 . . . . . . . . 9 ((𝑆 βˆ– (1...𝑁)) ∩ (1...𝑁)) = βˆ…
2119, 20sseqtrdi 4028 . . . . . . . 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 4394 . . . . . . . 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 6852 . . . . . . 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 838 . . . . . 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 6832 . . . . . 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 4149 . . . . . . 7 (((1...𝐴) βˆ– (1...𝑁)) βˆͺ (1...𝑁)) = ((1...𝑁) βˆͺ ((1...𝐴) βˆ– (1...𝑁)))
29 simplrr 777 . . . . . . . . 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 13559 . . . . . . . . 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 4477 . . . . . . . 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 2779 . . . . . 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 4130 . . . . . 6 (π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁)) β†’ ran π‘Ž βŠ† 𝑆)
3938adantl 481 . . . . 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 776 . . . . 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 4182 . . . 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 6793 . . . 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 583 . . 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 4197 . . . . . . . . 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 481 . . . . . . . . . . 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 4207 . . . . . . . . . 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 2783 . . . . . . . . 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 2779 . . . . . . . 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 2779 . . . . . . 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 5924 . . . . . . . 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 4160 . . . . 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 4149 . . . . . 6 (βˆ… βˆͺ ( I β†Ύ (1...𝑁))) = (( I β†Ύ (1...𝑁)) βˆͺ βˆ…)
61 un0 4386 . . . . . 6 (( I β†Ύ (1...𝑁)) βˆͺ βˆ…) = ( I β†Ύ (1...𝑁))
6260, 61eqtri 2755 . . . . 5 (βˆ… βˆͺ ( I β†Ύ (1...𝑁))) = ( I β†Ύ (1...𝑁))
6359, 62eqtrdi 2783 . . . 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 2779 . . 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 3473 . . . . 5 π‘Ž ∈ V
66 ovex 7447 . . . . . 6 (1...𝑁) ∈ V
67 resiexg 7912 . . . . . 6 ((1...𝑁) ∈ V β†’ ( I β†Ύ (1...𝑁)) ∈ V)
6866, 67ax-mp 5 . . . . 5 ( I β†Ύ (1...𝑁)) ∈ V
6965, 68unex 7740 . . . 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 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 2729 . . . . 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 3591 . . 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 583 . 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 1931 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 395   = wceq 1534  βˆƒwex 1774   ∈ wcel 2099  Vcvv 3469   βˆ– cdif 3941   βˆͺ cun 3942   ∩ cin 3943   βŠ† wss 3944  βˆ…c0 4318   I cid 5569  dom cdm 5672  ran crn 5673   β†Ύ cres 5674  Rel wrel 5677  β€“1-1β†’wf1 6539  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542  (class class class)co 7414  Fincfn 8953  1c1 11125  β„•0cn0 12488  β„€β‰₯cuz 12838  ...cfz 13502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7863  df-1st 7985  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-er 8716  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-card 9948  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-nn 12229  df-n0 12489  df-z 12575  df-uz 12839  df-fz 13503
This theorem is referenced by:  eldioph2b  42095
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