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Theorem eldioph2lem2 41127
Description: Lemma for eldioph2 41128. 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 13883 . . . 4 (1...𝑁) ∈ Fin
3 difinf 9263 . . . 4 ((Β¬ 𝑆 ∈ Fin ∧ (1...𝑁) ∈ Fin) β†’ Β¬ (𝑆 βˆ– (1...𝑁)) ∈ Fin)
41, 2, 3sylancl 587 . . 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 13883 . . . 4 (1...𝐴) ∈ Fin
6 diffi 9126 . . . 4 ((1...𝐴) ∈ Fin β†’ ((1...𝐴) βˆ– (1...𝑁)) ∈ Fin)
75, 6ax-mp 5 . . 3 ((1...𝐴) βˆ– (1...𝑁)) ∈ Fin
8 isinffi 9933 . . 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 587 . 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 6796 . . . . . . . 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 483 . . . . . . 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 6823 . . . . . . . 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 4433 . . . . . . . 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 6739 . . . . . . . . . . . 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 6677 . . . . . . . . . . 11 (π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁)) β†’ ran π‘Ž βŠ† (𝑆 βˆ– (1...𝑁)))
1817adantl 483 . . . . . . . . . 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 4196 . . . . . . . . 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 4433 . . . . . . . . 9 ((𝑆 βˆ– (1...𝑁)) ∩ (1...𝑁)) = βˆ…
2119, 20sseqtrdi 3995 . . . . . . . 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 4359 . . . . . . . 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 6804 . . . . . . 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 6784 . . . . . 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 4114 . . . . . . 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 13487 . . . . . . . . 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 4442 . . . . . . . 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 2785 . . . . . 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 6735 . . . . . 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 4095 . . . . . 6 (π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁)) β†’ ran π‘Ž βŠ† 𝑆)
3938adantl 483 . . . . 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 4147 . . . 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 6745 . . . 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 585 . . 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 5953 . . . 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 5960 . . . . . . . 8 dom (π‘Ž β†Ύ (1...𝑁)) = ((1...𝑁) ∩ dom π‘Ž)
46 incom 4162 . . . . . . . . 9 ((1...𝑁) ∩ dom π‘Ž) = (dom π‘Ž ∩ (1...𝑁))
47 f1dm 6743 . . . . . . . . . . . 12 (π‘Ž:((1...𝐴) βˆ– (1...𝑁))–1-1β†’(𝑆 βˆ– (1...𝑁)) β†’ dom π‘Ž = ((1...𝐴) βˆ– (1...𝑁)))
4847adantl 483 . . . . . . . . . . 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 4172 . . . . . . . . . 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 2789 . . . . . . . . 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 2785 . . . . . . . 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 2785 . . . . . . 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 5967 . . . . . . . 8 Rel (π‘Ž β†Ύ (1...𝑁))
54 reldm0 5884 . . . . . . . 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 5971 . . . . . . 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 4125 . . . . 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 4114 . . . . . 6 (βˆ… βˆͺ ( I β†Ύ (1...𝑁))) = (( I β†Ύ (1...𝑁)) βˆͺ βˆ…)
61 un0 4351 . . . . . 6 (( I β†Ύ (1...𝑁)) βˆͺ βˆ…) = ( I β†Ύ (1...𝑁))
6260, 61eqtri 2761 . . . . 5 (βˆ… βˆͺ ( I β†Ύ (1...𝑁))) = ( I β†Ύ (1...𝑁))
6359, 62eqtrdi 2789 . . . 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 2785 . . 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 3448 . . . . 5 π‘Ž ∈ V
66 ovex 7391 . . . . . 6 (1...𝑁) ∈ V
67 resiexg 7852 . . . . . 6 ((1...𝑁) ∈ V β†’ ( I β†Ύ (1...𝑁)) ∈ V)
6866, 67ax-mp 5 . . . . 5 ( I β†Ύ (1...𝑁)) ∈ V
6965, 68unex 7681 . . . 4 (π‘Ž βˆͺ ( I β†Ύ (1...𝑁))) ∈ V
70 f1eq1 6734 . . . . 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 5932 . . . . . 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 2735 . . . . 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 632 . . . 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 3564 . . 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 585 . 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 1939 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 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  Vcvv 3444   βˆ– cdif 3908   βˆͺ cun 3909   ∩ cin 3910   βŠ† wss 3911  βˆ…c0 4283   I cid 5531  dom cdm 5634  ran crn 5635   β†Ύ cres 5636  Rel wrel 5639  β€“1-1β†’wf1 6494  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497  (class class class)co 7358  Fincfn 8886  1c1 11057  β„•0cn0 12418  β„€β‰₯cuz 12768  ...cfz 13430
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 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-card 9880  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-n0 12419  df-z 12505  df-uz 12769  df-fz 13431
This theorem is referenced by:  eldioph2b  41129
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