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Theorem ctssdclemn0 7109
Description: Lemma for ctssdc 7112. The Β¬ βˆ… ∈ 𝑆 case. (Contributed by Jim Kingdon, 16-Aug-2023.)
Hypotheses
Ref Expression
ctssdclemn0.ss (πœ‘ β†’ 𝑆 βŠ† Ο‰)
ctssdclemn0.dc (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ DECID 𝑛 ∈ 𝑆)
ctssdclemn0.f (πœ‘ β†’ 𝐹:𝑆–onto→𝐴)
ctssdclemn0.n0 (πœ‘ β†’ Β¬ βˆ… ∈ 𝑆)
Assertion
Ref Expression
ctssdclemn0 (πœ‘ β†’ βˆƒπ‘” 𝑔:ω–ontoβ†’(𝐴 βŠ” 1o))
Distinct variable groups:   𝐴,𝑔   𝑔,𝐹   𝑆,𝑔   𝑆,𝑛
Allowed substitution hints:   πœ‘(𝑔,𝑛)   𝐴(𝑛)   𝐹(𝑛)

Proof of Theorem ctssdclemn0
Dummy variables π‘š π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ctssdclemn0.f . . . . . . . . 9 (πœ‘ β†’ 𝐹:𝑆–onto→𝐴)
21ad2antrr 488 . . . . . . . 8 (((πœ‘ ∧ π‘š ∈ Ο‰) ∧ π‘š ∈ 𝑆) β†’ 𝐹:𝑆–onto→𝐴)
3 fof 5439 . . . . . . . 8 (𝐹:𝑆–onto→𝐴 β†’ 𝐹:π‘†βŸΆπ΄)
42, 3syl 14 . . . . . . 7 (((πœ‘ ∧ π‘š ∈ Ο‰) ∧ π‘š ∈ 𝑆) β†’ 𝐹:π‘†βŸΆπ΄)
5 simpr 110 . . . . . . 7 (((πœ‘ ∧ π‘š ∈ Ο‰) ∧ π‘š ∈ 𝑆) β†’ π‘š ∈ 𝑆)
64, 5ffvelcdmd 5653 . . . . . 6 (((πœ‘ ∧ π‘š ∈ Ο‰) ∧ π‘š ∈ 𝑆) β†’ (πΉβ€˜π‘š) ∈ 𝐴)
7 djulcl 7050 . . . . . 6 ((πΉβ€˜π‘š) ∈ 𝐴 β†’ (inlβ€˜(πΉβ€˜π‘š)) ∈ (𝐴 βŠ” 1o))
86, 7syl 14 . . . . 5 (((πœ‘ ∧ π‘š ∈ Ο‰) ∧ π‘š ∈ 𝑆) β†’ (inlβ€˜(πΉβ€˜π‘š)) ∈ (𝐴 βŠ” 1o))
9 0lt1o 6441 . . . . . . 7 βˆ… ∈ 1o
10 djurcl 7051 . . . . . . 7 (βˆ… ∈ 1o β†’ (inrβ€˜βˆ…) ∈ (𝐴 βŠ” 1o))
119, 10ax-mp 5 . . . . . 6 (inrβ€˜βˆ…) ∈ (𝐴 βŠ” 1o)
1211a1i 9 . . . . 5 (((πœ‘ ∧ π‘š ∈ Ο‰) ∧ Β¬ π‘š ∈ 𝑆) β†’ (inrβ€˜βˆ…) ∈ (𝐴 βŠ” 1o))
13 eleq1 2240 . . . . . . 7 (𝑛 = π‘š β†’ (𝑛 ∈ 𝑆 ↔ π‘š ∈ 𝑆))
1413dcbid 838 . . . . . 6 (𝑛 = π‘š β†’ (DECID 𝑛 ∈ 𝑆 ↔ DECID π‘š ∈ 𝑆))
15 ctssdclemn0.dc . . . . . . 7 (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ DECID 𝑛 ∈ 𝑆)
1615adantr 276 . . . . . 6 ((πœ‘ ∧ π‘š ∈ Ο‰) β†’ βˆ€π‘› ∈ Ο‰ DECID 𝑛 ∈ 𝑆)
17 simpr 110 . . . . . 6 ((πœ‘ ∧ π‘š ∈ Ο‰) β†’ π‘š ∈ Ο‰)
1814, 16, 17rspcdva 2847 . . . . 5 ((πœ‘ ∧ π‘š ∈ Ο‰) β†’ DECID π‘š ∈ 𝑆)
198, 12, 18ifcldadc 3564 . . . 4 ((πœ‘ ∧ π‘š ∈ Ο‰) β†’ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…)) ∈ (𝐴 βŠ” 1o))
2019fmpttd 5672 . . 3 (πœ‘ β†’ (π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…))):Ο‰βŸΆ(𝐴 βŠ” 1o))
211ad3antrrr 492 . . . . . . . . 9 ((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 𝐴) ∧ π‘₯ = (inlβ€˜π‘§)) β†’ 𝐹:𝑆–onto→𝐴)
22 simplr 528 . . . . . . . . 9 ((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 𝐴) ∧ π‘₯ = (inlβ€˜π‘§)) β†’ 𝑧 ∈ 𝐴)
23 foelrn 5754 . . . . . . . . 9 ((𝐹:𝑆–onto→𝐴 ∧ 𝑧 ∈ 𝐴) β†’ βˆƒπ‘¦ ∈ 𝑆 𝑧 = (πΉβ€˜π‘¦))
2421, 22, 23syl2anc 411 . . . . . . . 8 ((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 𝐴) ∧ π‘₯ = (inlβ€˜π‘§)) β†’ βˆƒπ‘¦ ∈ 𝑆 𝑧 = (πΉβ€˜π‘¦))
25 simplr 528 . . . . . . . . . . . 12 ((((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 𝐴) ∧ π‘₯ = (inlβ€˜π‘§)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (πΉβ€˜π‘¦)) β†’ 𝑦 ∈ 𝑆)
2625iftrued 3542 . . . . . . . . . . 11 ((((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 𝐴) ∧ π‘₯ = (inlβ€˜π‘§)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (πΉβ€˜π‘¦)) β†’ if(𝑦 ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘¦)), (inrβ€˜βˆ…)) = (inlβ€˜(πΉβ€˜π‘¦)))
27 eqid 2177 . . . . . . . . . . . 12 (π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…))) = (π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…)))
28 eleq1 2240 . . . . . . . . . . . . 13 (π‘š = 𝑦 β†’ (π‘š ∈ 𝑆 ↔ 𝑦 ∈ 𝑆))
29 2fveq3 5521 . . . . . . . . . . . . 13 (π‘š = 𝑦 β†’ (inlβ€˜(πΉβ€˜π‘š)) = (inlβ€˜(πΉβ€˜π‘¦)))
3028, 29ifbieq1d 3557 . . . . . . . . . . . 12 (π‘š = 𝑦 β†’ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…)) = if(𝑦 ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘¦)), (inrβ€˜βˆ…)))
31 ctssdclemn0.ss . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝑆 βŠ† Ο‰)
3231ad5antr 496 . . . . . . . . . . . . 13 ((((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 𝐴) ∧ π‘₯ = (inlβ€˜π‘§)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (πΉβ€˜π‘¦)) β†’ 𝑆 βŠ† Ο‰)
3332, 25sseldd 3157 . . . . . . . . . . . 12 ((((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 𝐴) ∧ π‘₯ = (inlβ€˜π‘§)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (πΉβ€˜π‘¦)) β†’ 𝑦 ∈ Ο‰)
341, 3syl 14 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ 𝐹:π‘†βŸΆπ΄)
3534ad5antr 496 . . . . . . . . . . . . . . 15 ((((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 𝐴) ∧ π‘₯ = (inlβ€˜π‘§)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (πΉβ€˜π‘¦)) β†’ 𝐹:π‘†βŸΆπ΄)
3635, 25ffvelcdmd 5653 . . . . . . . . . . . . . 14 ((((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 𝐴) ∧ π‘₯ = (inlβ€˜π‘§)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (πΉβ€˜π‘¦)) β†’ (πΉβ€˜π‘¦) ∈ 𝐴)
37 djulcl 7050 . . . . . . . . . . . . . 14 ((πΉβ€˜π‘¦) ∈ 𝐴 β†’ (inlβ€˜(πΉβ€˜π‘¦)) ∈ (𝐴 βŠ” 1o))
3836, 37syl 14 . . . . . . . . . . . . 13 ((((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 𝐴) ∧ π‘₯ = (inlβ€˜π‘§)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (πΉβ€˜π‘¦)) β†’ (inlβ€˜(πΉβ€˜π‘¦)) ∈ (𝐴 βŠ” 1o))
3926, 38eqeltrd 2254 . . . . . . . . . . . 12 ((((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 𝐴) ∧ π‘₯ = (inlβ€˜π‘§)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (πΉβ€˜π‘¦)) β†’ if(𝑦 ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘¦)), (inrβ€˜βˆ…)) ∈ (𝐴 βŠ” 1o))
4027, 30, 33, 39fvmptd3 5610 . . . . . . . . . . 11 ((((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 𝐴) ∧ π‘₯ = (inlβ€˜π‘§)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (πΉβ€˜π‘¦)) β†’ ((π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…)))β€˜π‘¦) = if(𝑦 ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘¦)), (inrβ€˜βˆ…)))
41 simpllr 534 . . . . . . . . . . . 12 ((((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 𝐴) ∧ π‘₯ = (inlβ€˜π‘§)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (πΉβ€˜π‘¦)) β†’ π‘₯ = (inlβ€˜π‘§))
42 simpr 110 . . . . . . . . . . . . 13 ((((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 𝐴) ∧ π‘₯ = (inlβ€˜π‘§)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (πΉβ€˜π‘¦)) β†’ 𝑧 = (πΉβ€˜π‘¦))
4342fveq2d 5520 . . . . . . . . . . . 12 ((((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 𝐴) ∧ π‘₯ = (inlβ€˜π‘§)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (πΉβ€˜π‘¦)) β†’ (inlβ€˜π‘§) = (inlβ€˜(πΉβ€˜π‘¦)))
4441, 43eqtrd 2210 . . . . . . . . . . 11 ((((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 𝐴) ∧ π‘₯ = (inlβ€˜π‘§)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (πΉβ€˜π‘¦)) β†’ π‘₯ = (inlβ€˜(πΉβ€˜π‘¦)))
4526, 40, 443eqtr4rd 2221 . . . . . . . . . 10 ((((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 𝐴) ∧ π‘₯ = (inlβ€˜π‘§)) ∧ 𝑦 ∈ 𝑆) ∧ 𝑧 = (πΉβ€˜π‘¦)) β†’ π‘₯ = ((π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…)))β€˜π‘¦))
4645ex 115 . . . . . . . . 9 (((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 𝐴) ∧ π‘₯ = (inlβ€˜π‘§)) ∧ 𝑦 ∈ 𝑆) β†’ (𝑧 = (πΉβ€˜π‘¦) β†’ π‘₯ = ((π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…)))β€˜π‘¦)))
4746reximdva 2579 . . . . . . . 8 ((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 𝐴) ∧ π‘₯ = (inlβ€˜π‘§)) β†’ (βˆƒπ‘¦ ∈ 𝑆 𝑧 = (πΉβ€˜π‘¦) β†’ βˆƒπ‘¦ ∈ 𝑆 π‘₯ = ((π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…)))β€˜π‘¦)))
4824, 47mpd 13 . . . . . . 7 ((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 𝐴) ∧ π‘₯ = (inlβ€˜π‘§)) β†’ βˆƒπ‘¦ ∈ 𝑆 π‘₯ = ((π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…)))β€˜π‘¦))
49 ssrexv 3221 . . . . . . . . 9 (𝑆 βŠ† Ο‰ β†’ (βˆƒπ‘¦ ∈ 𝑆 π‘₯ = ((π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…)))β€˜π‘¦) β†’ βˆƒπ‘¦ ∈ Ο‰ π‘₯ = ((π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…)))β€˜π‘¦)))
5031, 49syl 14 . . . . . . . 8 (πœ‘ β†’ (βˆƒπ‘¦ ∈ 𝑆 π‘₯ = ((π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…)))β€˜π‘¦) β†’ βˆƒπ‘¦ ∈ Ο‰ π‘₯ = ((π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…)))β€˜π‘¦)))
5150ad3antrrr 492 . . . . . . 7 ((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 𝐴) ∧ π‘₯ = (inlβ€˜π‘§)) β†’ (βˆƒπ‘¦ ∈ 𝑆 π‘₯ = ((π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…)))β€˜π‘¦) β†’ βˆƒπ‘¦ ∈ Ο‰ π‘₯ = ((π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…)))β€˜π‘¦)))
5248, 51mpd 13 . . . . . 6 ((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 𝐴) ∧ π‘₯ = (inlβ€˜π‘§)) β†’ βˆƒπ‘¦ ∈ Ο‰ π‘₯ = ((π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…)))β€˜π‘¦))
5352rexlimdva2 2597 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) β†’ (βˆƒπ‘§ ∈ 𝐴 π‘₯ = (inlβ€˜π‘§) β†’ βˆƒπ‘¦ ∈ Ο‰ π‘₯ = ((π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…)))β€˜π‘¦)))
54 peano1 4594 . . . . . . . 8 βˆ… ∈ Ο‰
5554a1i 9 . . . . . . 7 ((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 1o) ∧ π‘₯ = (inrβ€˜π‘§)) β†’ βˆ… ∈ Ο‰)
56 ctssdclemn0.n0 . . . . . . . . . 10 (πœ‘ β†’ Β¬ βˆ… ∈ 𝑆)
5756ad3antrrr 492 . . . . . . . . 9 ((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 1o) ∧ π‘₯ = (inrβ€˜π‘§)) β†’ Β¬ βˆ… ∈ 𝑆)
5857iffalsed 3545 . . . . . . . 8 ((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 1o) ∧ π‘₯ = (inrβ€˜π‘§)) β†’ if(βˆ… ∈ 𝑆, (inlβ€˜(πΉβ€˜βˆ…)), (inrβ€˜βˆ…)) = (inrβ€˜βˆ…))
59 eleq1 2240 . . . . . . . . . 10 (π‘š = βˆ… β†’ (π‘š ∈ 𝑆 ↔ βˆ… ∈ 𝑆))
60 2fveq3 5521 . . . . . . . . . 10 (π‘š = βˆ… β†’ (inlβ€˜(πΉβ€˜π‘š)) = (inlβ€˜(πΉβ€˜βˆ…)))
6159, 60ifbieq1d 3557 . . . . . . . . 9 (π‘š = βˆ… β†’ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…)) = if(βˆ… ∈ 𝑆, (inlβ€˜(πΉβ€˜βˆ…)), (inrβ€˜βˆ…)))
6258, 11eqeltrdi 2268 . . . . . . . . 9 ((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 1o) ∧ π‘₯ = (inrβ€˜π‘§)) β†’ if(βˆ… ∈ 𝑆, (inlβ€˜(πΉβ€˜βˆ…)), (inrβ€˜βˆ…)) ∈ (𝐴 βŠ” 1o))
6327, 61, 55, 62fvmptd3 5610 . . . . . . . 8 ((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 1o) ∧ π‘₯ = (inrβ€˜π‘§)) β†’ ((π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…)))β€˜βˆ…) = if(βˆ… ∈ 𝑆, (inlβ€˜(πΉβ€˜βˆ…)), (inrβ€˜βˆ…)))
64 simpr 110 . . . . . . . . 9 ((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 1o) ∧ π‘₯ = (inrβ€˜π‘§)) β†’ π‘₯ = (inrβ€˜π‘§))
65 simplr 528 . . . . . . . . . . 11 ((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 1o) ∧ π‘₯ = (inrβ€˜π‘§)) β†’ 𝑧 ∈ 1o)
66 el1o 6438 . . . . . . . . . . 11 (𝑧 ∈ 1o ↔ 𝑧 = βˆ…)
6765, 66sylib 122 . . . . . . . . . 10 ((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 1o) ∧ π‘₯ = (inrβ€˜π‘§)) β†’ 𝑧 = βˆ…)
6867fveq2d 5520 . . . . . . . . 9 ((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 1o) ∧ π‘₯ = (inrβ€˜π‘§)) β†’ (inrβ€˜π‘§) = (inrβ€˜βˆ…))
6964, 68eqtrd 2210 . . . . . . . 8 ((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 1o) ∧ π‘₯ = (inrβ€˜π‘§)) β†’ π‘₯ = (inrβ€˜βˆ…))
7058, 63, 693eqtr4rd 2221 . . . . . . 7 ((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 1o) ∧ π‘₯ = (inrβ€˜π‘§)) β†’ π‘₯ = ((π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…)))β€˜βˆ…))
71 fveq2 5516 . . . . . . . 8 (𝑦 = βˆ… β†’ ((π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…)))β€˜π‘¦) = ((π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…)))β€˜βˆ…))
7271rspceeqv 2860 . . . . . . 7 ((βˆ… ∈ Ο‰ ∧ π‘₯ = ((π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…)))β€˜βˆ…)) β†’ βˆƒπ‘¦ ∈ Ο‰ π‘₯ = ((π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…)))β€˜π‘¦))
7355, 70, 72syl2anc 411 . . . . . 6 ((((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) ∧ 𝑧 ∈ 1o) ∧ π‘₯ = (inrβ€˜π‘§)) β†’ βˆƒπ‘¦ ∈ Ο‰ π‘₯ = ((π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…)))β€˜π‘¦))
7473rexlimdva2 2597 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) β†’ (βˆƒπ‘§ ∈ 1o π‘₯ = (inrβ€˜π‘§) β†’ βˆƒπ‘¦ ∈ Ο‰ π‘₯ = ((π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…)))β€˜π‘¦)))
75 djur 7068 . . . . . . 7 (π‘₯ ∈ (𝐴 βŠ” 1o) ↔ (βˆƒπ‘§ ∈ 𝐴 π‘₯ = (inlβ€˜π‘§) ∨ βˆƒπ‘§ ∈ 1o π‘₯ = (inrβ€˜π‘§)))
7675biimpi 120 . . . . . 6 (π‘₯ ∈ (𝐴 βŠ” 1o) β†’ (βˆƒπ‘§ ∈ 𝐴 π‘₯ = (inlβ€˜π‘§) ∨ βˆƒπ‘§ ∈ 1o π‘₯ = (inrβ€˜π‘§)))
7776adantl 277 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) β†’ (βˆƒπ‘§ ∈ 𝐴 π‘₯ = (inlβ€˜π‘§) ∨ βˆƒπ‘§ ∈ 1o π‘₯ = (inrβ€˜π‘§)))
7853, 74, 77mpjaod 718 . . . 4 ((πœ‘ ∧ π‘₯ ∈ (𝐴 βŠ” 1o)) β†’ βˆƒπ‘¦ ∈ Ο‰ π‘₯ = ((π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…)))β€˜π‘¦))
7978ralrimiva 2550 . . 3 (πœ‘ β†’ βˆ€π‘₯ ∈ (𝐴 βŠ” 1o)βˆƒπ‘¦ ∈ Ο‰ π‘₯ = ((π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…)))β€˜π‘¦))
80 dffo3 5664 . . 3 ((π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…))):ω–ontoβ†’(𝐴 βŠ” 1o) ↔ ((π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…))):Ο‰βŸΆ(𝐴 βŠ” 1o) ∧ βˆ€π‘₯ ∈ (𝐴 βŠ” 1o)βˆƒπ‘¦ ∈ Ο‰ π‘₯ = ((π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…)))β€˜π‘¦)))
8120, 79, 80sylanbrc 417 . 2 (πœ‘ β†’ (π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…))):ω–ontoβ†’(𝐴 βŠ” 1o))
82 omex 4593 . . . 4 Ο‰ ∈ V
8382mptex 5743 . . 3 (π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…))) ∈ V
84 foeq1 5435 . . 3 (𝑔 = (π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…))) β†’ (𝑔:ω–ontoβ†’(𝐴 βŠ” 1o) ↔ (π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…))):ω–ontoβ†’(𝐴 βŠ” 1o)))
8583, 84spcev 2833 . 2 ((π‘š ∈ Ο‰ ↦ if(π‘š ∈ 𝑆, (inlβ€˜(πΉβ€˜π‘š)), (inrβ€˜βˆ…))):ω–ontoβ†’(𝐴 βŠ” 1o) β†’ βˆƒπ‘” 𝑔:ω–ontoβ†’(𝐴 βŠ” 1o))
8681, 85syl 14 1 (πœ‘ β†’ βˆƒπ‘” 𝑔:ω–ontoβ†’(𝐴 βŠ” 1o))
Colors of variables: wff set class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 104   ∨ wo 708  DECID wdc 834   = wceq 1353  βˆƒwex 1492   ∈ wcel 2148  βˆ€wral 2455  βˆƒwrex 2456   βŠ† wss 3130  βˆ…c0 3423  ifcif 3535   ↦ cmpt 4065  Ο‰com 4590  βŸΆwf 5213  β€“ontoβ†’wfo 5215  β€˜cfv 5217  1oc1o 6410   βŠ” cdju 7036  inlcinl 7044  inrcinr 7045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-1st 6141  df-2nd 6142  df-1o 6417  df-dju 7037  df-inl 7046  df-inr 7047
This theorem is referenced by:  ctssdc  7112
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