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Theorem ctmlemr 7106
Description: Lemma for ctm 7107. One of the directions of the biconditional. (Contributed by Jim Kingdon, 16-Mar-2023.)
Assertion
Ref Expression
ctmlemr (βˆƒπ‘₯ π‘₯ ∈ 𝐴 β†’ (βˆƒπ‘“ 𝑓:ω–onto→𝐴 β†’ βˆƒπ‘“ 𝑓:ω–ontoβ†’(𝐴 βŠ” 1o)))
Distinct variable groups:   𝐴,𝑓   π‘₯,𝑓
Allowed substitution hint:   𝐴(π‘₯)

Proof of Theorem ctmlemr
Dummy variables 𝑔 𝑛 𝑒 𝑀 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0lt1o 6440 . . . . . . . . . 10 βˆ… ∈ 1o
2 djurcl 7050 . . . . . . . . . 10 (βˆ… ∈ 1o β†’ (inrβ€˜βˆ…) ∈ (𝐴 βŠ” 1o))
31, 2ax-mp 5 . . . . . . . . 9 (inrβ€˜βˆ…) ∈ (𝐴 βŠ” 1o)
43a1i 9 . . . . . . . 8 ((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑛 ∈ Ο‰) ∧ 𝑛 = βˆ…) β†’ (inrβ€˜βˆ…) ∈ (𝐴 βŠ” 1o))
5 simpllr 534 . . . . . . . . . . 11 ((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑛 ∈ Ο‰) ∧ Β¬ 𝑛 = βˆ…) β†’ 𝑓:ω–onto→𝐴)
6 fof 5438 . . . . . . . . . . 11 (𝑓:ω–onto→𝐴 β†’ 𝑓:Ο‰βŸΆπ΄)
75, 6syl 14 . . . . . . . . . 10 ((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑛 ∈ Ο‰) ∧ Β¬ 𝑛 = βˆ…) β†’ 𝑓:Ο‰βŸΆπ΄)
8 nnpredcl 4622 . . . . . . . . . . 11 (𝑛 ∈ Ο‰ β†’ βˆͺ 𝑛 ∈ Ο‰)
98ad2antlr 489 . . . . . . . . . 10 ((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑛 ∈ Ο‰) ∧ Β¬ 𝑛 = βˆ…) β†’ βˆͺ 𝑛 ∈ Ο‰)
107, 9ffvelcdmd 5652 . . . . . . . . 9 ((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑛 ∈ Ο‰) ∧ Β¬ 𝑛 = βˆ…) β†’ (π‘“β€˜βˆͺ 𝑛) ∈ 𝐴)
11 djulcl 7049 . . . . . . . . 9 ((π‘“β€˜βˆͺ 𝑛) ∈ 𝐴 β†’ (inlβ€˜(π‘“β€˜βˆͺ 𝑛)) ∈ (𝐴 βŠ” 1o))
1210, 11syl 14 . . . . . . . 8 ((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑛 ∈ Ο‰) ∧ Β¬ 𝑛 = βˆ…) β†’ (inlβ€˜(π‘“β€˜βˆͺ 𝑛)) ∈ (𝐴 βŠ” 1o))
13 nndceq0 4617 . . . . . . . . 9 (𝑛 ∈ Ο‰ β†’ DECID 𝑛 = βˆ…)
1413adantl 277 . . . . . . . 8 (((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑛 ∈ Ο‰) β†’ DECID 𝑛 = βˆ…)
154, 12, 14ifcldadc 3563 . . . . . . 7 (((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑛 ∈ Ο‰) β†’ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛))) ∈ (𝐴 βŠ” 1o))
1615fmpttd 5671 . . . . . 6 ((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) β†’ (𝑛 ∈ Ο‰ ↦ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛)))):Ο‰βŸΆ(𝐴 βŠ” 1o))
17 simpllr 534 . . . . . . . . . . 11 ((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘€))) β†’ 𝑓:ω–onto→𝐴)
18 simprl 529 . . . . . . . . . . 11 ((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘€))) β†’ 𝑀 ∈ 𝐴)
19 foelrn 5753 . . . . . . . . . . 11 ((𝑓:ω–onto→𝐴 ∧ 𝑀 ∈ 𝐴) β†’ βˆƒπ‘’ ∈ Ο‰ 𝑀 = (π‘“β€˜π‘’))
2017, 18, 19syl2anc 411 . . . . . . . . . 10 ((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘€))) β†’ βˆƒπ‘’ ∈ Ο‰ 𝑀 = (π‘“β€˜π‘’))
21 peano2 4594 . . . . . . . . . . . 12 (𝑒 ∈ Ο‰ β†’ suc 𝑒 ∈ Ο‰)
2221ad2antrl 490 . . . . . . . . . . 11 (((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘€))) ∧ (𝑒 ∈ Ο‰ ∧ 𝑀 = (π‘“β€˜π‘’))) β†’ suc 𝑒 ∈ Ο‰)
23 simplrr 536 . . . . . . . . . . . . . 14 (((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘€))) ∧ (𝑒 ∈ Ο‰ ∧ 𝑀 = (π‘“β€˜π‘’))) β†’ 𝑦 = (inlβ€˜π‘€))
24 simprl 529 . . . . . . . . . . . . . . . . . . 19 (((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘€))) ∧ (𝑒 ∈ Ο‰ ∧ 𝑀 = (π‘“β€˜π‘’))) β†’ 𝑒 ∈ Ο‰)
25 nnord 4611 . . . . . . . . . . . . . . . . . . 19 (𝑒 ∈ Ο‰ β†’ Ord 𝑒)
26 ordtr 4378 . . . . . . . . . . . . . . . . . . 19 (Ord 𝑒 β†’ Tr 𝑒)
2724, 25, 263syl 17 . . . . . . . . . . . . . . . . . 18 (((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘€))) ∧ (𝑒 ∈ Ο‰ ∧ 𝑀 = (π‘“β€˜π‘’))) β†’ Tr 𝑒)
28 unisucg 4414 . . . . . . . . . . . . . . . . . . 19 (𝑒 ∈ Ο‰ β†’ (Tr 𝑒 ↔ βˆͺ suc 𝑒 = 𝑒))
2928ad2antrl 490 . . . . . . . . . . . . . . . . . 18 (((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘€))) ∧ (𝑒 ∈ Ο‰ ∧ 𝑀 = (π‘“β€˜π‘’))) β†’ (Tr 𝑒 ↔ βˆͺ suc 𝑒 = 𝑒))
3027, 29mpbid 147 . . . . . . . . . . . . . . . . 17 (((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘€))) ∧ (𝑒 ∈ Ο‰ ∧ 𝑀 = (π‘“β€˜π‘’))) β†’ βˆͺ suc 𝑒 = 𝑒)
3130fveq2d 5519 . . . . . . . . . . . . . . . 16 (((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘€))) ∧ (𝑒 ∈ Ο‰ ∧ 𝑀 = (π‘“β€˜π‘’))) β†’ (π‘“β€˜βˆͺ suc 𝑒) = (π‘“β€˜π‘’))
32 simprr 531 . . . . . . . . . . . . . . . 16 (((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘€))) ∧ (𝑒 ∈ Ο‰ ∧ 𝑀 = (π‘“β€˜π‘’))) β†’ 𝑀 = (π‘“β€˜π‘’))
3331, 32eqtr4d 2213 . . . . . . . . . . . . . . 15 (((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘€))) ∧ (𝑒 ∈ Ο‰ ∧ 𝑀 = (π‘“β€˜π‘’))) β†’ (π‘“β€˜βˆͺ suc 𝑒) = 𝑀)
3433fveq2d 5519 . . . . . . . . . . . . . 14 (((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘€))) ∧ (𝑒 ∈ Ο‰ ∧ 𝑀 = (π‘“β€˜π‘’))) β†’ (inlβ€˜(π‘“β€˜βˆͺ suc 𝑒)) = (inlβ€˜π‘€))
3523, 34eqtr4d 2213 . . . . . . . . . . . . 13 (((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘€))) ∧ (𝑒 ∈ Ο‰ ∧ 𝑀 = (π‘“β€˜π‘’))) β†’ 𝑦 = (inlβ€˜(π‘“β€˜βˆͺ suc 𝑒)))
36 peano3 4595 . . . . . . . . . . . . . . . 16 (𝑒 ∈ Ο‰ β†’ suc 𝑒 β‰  βˆ…)
3736neneqd 2368 . . . . . . . . . . . . . . 15 (𝑒 ∈ Ο‰ β†’ Β¬ suc 𝑒 = βˆ…)
3837ad2antrl 490 . . . . . . . . . . . . . 14 (((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘€))) ∧ (𝑒 ∈ Ο‰ ∧ 𝑀 = (π‘“β€˜π‘’))) β†’ Β¬ suc 𝑒 = βˆ…)
3938iffalsed 3544 . . . . . . . . . . . . 13 (((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘€))) ∧ (𝑒 ∈ Ο‰ ∧ 𝑀 = (π‘“β€˜π‘’))) β†’ if(suc 𝑒 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ suc 𝑒))) = (inlβ€˜(π‘“β€˜βˆͺ suc 𝑒)))
4035, 39eqtr4d 2213 . . . . . . . . . . . 12 (((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘€))) ∧ (𝑒 ∈ Ο‰ ∧ 𝑀 = (π‘“β€˜π‘’))) β†’ 𝑦 = if(suc 𝑒 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ suc 𝑒))))
41 eqid 2177 . . . . . . . . . . . . 13 (𝑛 ∈ Ο‰ ↦ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛)))) = (𝑛 ∈ Ο‰ ↦ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛))))
42 eqeq1 2184 . . . . . . . . . . . . . 14 (𝑛 = suc 𝑒 β†’ (𝑛 = βˆ… ↔ suc 𝑒 = βˆ…))
43 unieq 3818 . . . . . . . . . . . . . . . 16 (𝑛 = suc 𝑒 β†’ βˆͺ 𝑛 = βˆͺ suc 𝑒)
4443fveq2d 5519 . . . . . . . . . . . . . . 15 (𝑛 = suc 𝑒 β†’ (π‘“β€˜βˆͺ 𝑛) = (π‘“β€˜βˆͺ suc 𝑒))
4544fveq2d 5519 . . . . . . . . . . . . . 14 (𝑛 = suc 𝑒 β†’ (inlβ€˜(π‘“β€˜βˆͺ 𝑛)) = (inlβ€˜(π‘“β€˜βˆͺ suc 𝑒)))
4642, 45ifbieq2d 3558 . . . . . . . . . . . . 13 (𝑛 = suc 𝑒 β†’ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛))) = if(suc 𝑒 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ suc 𝑒))))
47 simpllr 534 . . . . . . . . . . . . . 14 (((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘€))) ∧ (𝑒 ∈ Ο‰ ∧ 𝑀 = (π‘“β€˜π‘’))) β†’ 𝑦 ∈ (𝐴 βŠ” 1o))
4840, 47eqeltrrd 2255 . . . . . . . . . . . . 13 (((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘€))) ∧ (𝑒 ∈ Ο‰ ∧ 𝑀 = (π‘“β€˜π‘’))) β†’ if(suc 𝑒 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ suc 𝑒))) ∈ (𝐴 βŠ” 1o))
4941, 46, 22, 48fvmptd3 5609 . . . . . . . . . . . 12 (((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘€))) ∧ (𝑒 ∈ Ο‰ ∧ 𝑀 = (π‘“β€˜π‘’))) β†’ ((𝑛 ∈ Ο‰ ↦ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛))))β€˜suc 𝑒) = if(suc 𝑒 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ suc 𝑒))))
5040, 49eqtr4d 2213 . . . . . . . . . . 11 (((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘€))) ∧ (𝑒 ∈ Ο‰ ∧ 𝑀 = (π‘“β€˜π‘’))) β†’ 𝑦 = ((𝑛 ∈ Ο‰ ↦ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛))))β€˜suc 𝑒))
51 fveq2 5515 . . . . . . . . . . . 12 (𝑧 = suc 𝑒 β†’ ((𝑛 ∈ Ο‰ ↦ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛))))β€˜π‘§) = ((𝑛 ∈ Ο‰ ↦ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛))))β€˜suc 𝑒))
5251rspceeqv 2859 . . . . . . . . . . 11 ((suc 𝑒 ∈ Ο‰ ∧ 𝑦 = ((𝑛 ∈ Ο‰ ↦ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛))))β€˜suc 𝑒)) β†’ βˆƒπ‘§ ∈ Ο‰ 𝑦 = ((𝑛 ∈ Ο‰ ↦ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛))))β€˜π‘§))
5322, 50, 52syl2anc 411 . . . . . . . . . 10 (((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘€))) ∧ (𝑒 ∈ Ο‰ ∧ 𝑀 = (π‘“β€˜π‘’))) β†’ βˆƒπ‘§ ∈ Ο‰ 𝑦 = ((𝑛 ∈ Ο‰ ↦ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛))))β€˜π‘§))
5420, 53rexlimddv 2599 . . . . . . . . 9 ((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 𝐴 ∧ 𝑦 = (inlβ€˜π‘€))) β†’ βˆƒπ‘§ ∈ Ο‰ 𝑦 = ((𝑛 ∈ Ο‰ ↦ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛))))β€˜π‘§))
5554rexlimdvaa 2595 . . . . . . . 8 (((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) β†’ (βˆƒπ‘€ ∈ 𝐴 𝑦 = (inlβ€˜π‘€) β†’ βˆƒπ‘§ ∈ Ο‰ 𝑦 = ((𝑛 ∈ Ο‰ ↦ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛))))β€˜π‘§)))
56 peano1 4593 . . . . . . . . . 10 βˆ… ∈ Ο‰
57 simprr 531 . . . . . . . . . . . . 13 ((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 1o ∧ 𝑦 = (inrβ€˜π‘€))) β†’ 𝑦 = (inrβ€˜π‘€))
58 simprl 529 . . . . . . . . . . . . . . 15 ((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 1o ∧ 𝑦 = (inrβ€˜π‘€))) β†’ 𝑀 ∈ 1o)
59 el1o 6437 . . . . . . . . . . . . . . 15 (𝑀 ∈ 1o ↔ 𝑀 = βˆ…)
6058, 59sylib 122 . . . . . . . . . . . . . 14 ((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 1o ∧ 𝑦 = (inrβ€˜π‘€))) β†’ 𝑀 = βˆ…)
6160fveq2d 5519 . . . . . . . . . . . . 13 ((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 1o ∧ 𝑦 = (inrβ€˜π‘€))) β†’ (inrβ€˜π‘€) = (inrβ€˜βˆ…))
6257, 61eqtrd 2210 . . . . . . . . . . . 12 ((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 1o ∧ 𝑦 = (inrβ€˜π‘€))) β†’ 𝑦 = (inrβ€˜βˆ…))
63 eqid 2177 . . . . . . . . . . . . 13 βˆ… = βˆ…
6463iftruei 3540 . . . . . . . . . . . 12 if(βˆ… = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ βˆ…))) = (inrβ€˜βˆ…)
6562, 64eqtr4di 2228 . . . . . . . . . . 11 ((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 1o ∧ 𝑦 = (inrβ€˜π‘€))) β†’ 𝑦 = if(βˆ… = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ βˆ…))))
6664, 3eqeltri 2250 . . . . . . . . . . . . 13 if(βˆ… = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ βˆ…))) ∈ (𝐴 βŠ” 1o)
6766a1i 9 . . . . . . . . . . . 12 ((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 1o ∧ 𝑦 = (inrβ€˜π‘€))) β†’ if(βˆ… = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ βˆ…))) ∈ (𝐴 βŠ” 1o))
68 eqeq1 2184 . . . . . . . . . . . . . 14 (𝑛 = βˆ… β†’ (𝑛 = βˆ… ↔ βˆ… = βˆ…))
69 unieq 3818 . . . . . . . . . . . . . . . 16 (𝑛 = βˆ… β†’ βˆͺ 𝑛 = βˆͺ βˆ…)
7069fveq2d 5519 . . . . . . . . . . . . . . 15 (𝑛 = βˆ… β†’ (π‘“β€˜βˆͺ 𝑛) = (π‘“β€˜βˆͺ βˆ…))
7170fveq2d 5519 . . . . . . . . . . . . . 14 (𝑛 = βˆ… β†’ (inlβ€˜(π‘“β€˜βˆͺ 𝑛)) = (inlβ€˜(π‘“β€˜βˆͺ βˆ…)))
7268, 71ifbieq2d 3558 . . . . . . . . . . . . 13 (𝑛 = βˆ… β†’ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛))) = if(βˆ… = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ βˆ…))))
7372, 41fvmptg 5592 . . . . . . . . . . . 12 ((βˆ… ∈ Ο‰ ∧ if(βˆ… = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ βˆ…))) ∈ (𝐴 βŠ” 1o)) β†’ ((𝑛 ∈ Ο‰ ↦ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛))))β€˜βˆ…) = if(βˆ… = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ βˆ…))))
7456, 67, 73sylancr 414 . . . . . . . . . . 11 ((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 1o ∧ 𝑦 = (inrβ€˜π‘€))) β†’ ((𝑛 ∈ Ο‰ ↦ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛))))β€˜βˆ…) = if(βˆ… = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ βˆ…))))
7565, 74eqtr4d 2213 . . . . . . . . . 10 ((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 1o ∧ 𝑦 = (inrβ€˜π‘€))) β†’ 𝑦 = ((𝑛 ∈ Ο‰ ↦ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛))))β€˜βˆ…))
76 fveq2 5515 . . . . . . . . . . 11 (𝑧 = βˆ… β†’ ((𝑛 ∈ Ο‰ ↦ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛))))β€˜π‘§) = ((𝑛 ∈ Ο‰ ↦ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛))))β€˜βˆ…))
7776rspceeqv 2859 . . . . . . . . . 10 ((βˆ… ∈ Ο‰ ∧ 𝑦 = ((𝑛 ∈ Ο‰ ↦ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛))))β€˜βˆ…)) β†’ βˆƒπ‘§ ∈ Ο‰ 𝑦 = ((𝑛 ∈ Ο‰ ↦ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛))))β€˜π‘§))
7856, 75, 77sylancr 414 . . . . . . . . 9 ((((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) ∧ (𝑀 ∈ 1o ∧ 𝑦 = (inrβ€˜π‘€))) β†’ βˆƒπ‘§ ∈ Ο‰ 𝑦 = ((𝑛 ∈ Ο‰ ↦ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛))))β€˜π‘§))
7978rexlimdvaa 2595 . . . . . . . 8 (((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) β†’ (βˆƒπ‘€ ∈ 1o 𝑦 = (inrβ€˜π‘€) β†’ βˆƒπ‘§ ∈ Ο‰ 𝑦 = ((𝑛 ∈ Ο‰ ↦ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛))))β€˜π‘§)))
80 djur 7067 . . . . . . . . . 10 (𝑦 ∈ (𝐴 βŠ” 1o) ↔ (βˆƒπ‘€ ∈ 𝐴 𝑦 = (inlβ€˜π‘€) ∨ βˆƒπ‘€ ∈ 1o 𝑦 = (inrβ€˜π‘€)))
8180biimpi 120 . . . . . . . . 9 (𝑦 ∈ (𝐴 βŠ” 1o) β†’ (βˆƒπ‘€ ∈ 𝐴 𝑦 = (inlβ€˜π‘€) ∨ βˆƒπ‘€ ∈ 1o 𝑦 = (inrβ€˜π‘€)))
8281adantl 277 . . . . . . . 8 (((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) β†’ (βˆƒπ‘€ ∈ 𝐴 𝑦 = (inlβ€˜π‘€) ∨ βˆƒπ‘€ ∈ 1o 𝑦 = (inrβ€˜π‘€)))
8355, 79, 82mpjaod 718 . . . . . . 7 (((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) ∧ 𝑦 ∈ (𝐴 βŠ” 1o)) β†’ βˆƒπ‘§ ∈ Ο‰ 𝑦 = ((𝑛 ∈ Ο‰ ↦ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛))))β€˜π‘§))
8483ralrimiva 2550 . . . . . 6 ((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) β†’ βˆ€π‘¦ ∈ (𝐴 βŠ” 1o)βˆƒπ‘§ ∈ Ο‰ 𝑦 = ((𝑛 ∈ Ο‰ ↦ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛))))β€˜π‘§))
85 dffo3 5663 . . . . . 6 ((𝑛 ∈ Ο‰ ↦ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛)))):ω–ontoβ†’(𝐴 βŠ” 1o) ↔ ((𝑛 ∈ Ο‰ ↦ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛)))):Ο‰βŸΆ(𝐴 βŠ” 1o) ∧ βˆ€π‘¦ ∈ (𝐴 βŠ” 1o)βˆƒπ‘§ ∈ Ο‰ 𝑦 = ((𝑛 ∈ Ο‰ ↦ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛))))β€˜π‘§)))
8616, 84, 85sylanbrc 417 . . . . 5 ((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) β†’ (𝑛 ∈ Ο‰ ↦ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛)))):ω–ontoβ†’(𝐴 βŠ” 1o))
87 omex 4592 . . . . . . 7 Ο‰ ∈ V
8887mptex 5742 . . . . . 6 (𝑛 ∈ Ο‰ ↦ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛)))) ∈ V
89 foeq1 5434 . . . . . 6 (𝑔 = (𝑛 ∈ Ο‰ ↦ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛)))) β†’ (𝑔:ω–ontoβ†’(𝐴 βŠ” 1o) ↔ (𝑛 ∈ Ο‰ ↦ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛)))):ω–ontoβ†’(𝐴 βŠ” 1o)))
9088, 89spcev 2832 . . . . 5 ((𝑛 ∈ Ο‰ ↦ if(𝑛 = βˆ…, (inrβ€˜βˆ…), (inlβ€˜(π‘“β€˜βˆͺ 𝑛)))):ω–ontoβ†’(𝐴 βŠ” 1o) β†’ βˆƒπ‘” 𝑔:ω–ontoβ†’(𝐴 βŠ” 1o))
9186, 90syl 14 . . . 4 ((βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∧ 𝑓:ω–onto→𝐴) β†’ βˆƒπ‘” 𝑔:ω–ontoβ†’(𝐴 βŠ” 1o))
9291ex 115 . . 3 (βˆƒπ‘₯ π‘₯ ∈ 𝐴 β†’ (𝑓:ω–onto→𝐴 β†’ βˆƒπ‘” 𝑔:ω–ontoβ†’(𝐴 βŠ” 1o)))
9392exlimdv 1819 . 2 (βˆƒπ‘₯ π‘₯ ∈ 𝐴 β†’ (βˆƒπ‘“ 𝑓:ω–onto→𝐴 β†’ βˆƒπ‘” 𝑔:ω–ontoβ†’(𝐴 βŠ” 1o)))
94 foeq1 5434 . . 3 (𝑓 = 𝑔 β†’ (𝑓:ω–ontoβ†’(𝐴 βŠ” 1o) ↔ 𝑔:ω–ontoβ†’(𝐴 βŠ” 1o)))
9594cbvexv 1918 . 2 (βˆƒπ‘“ 𝑓:ω–ontoβ†’(𝐴 βŠ” 1o) ↔ βˆƒπ‘” 𝑔:ω–ontoβ†’(𝐴 βŠ” 1o))
9693, 95imbitrrdi 162 1 (βˆƒπ‘₯ π‘₯ ∈ 𝐴 β†’ (βˆƒπ‘“ 𝑓:ω–onto→𝐴 β†’ βˆƒπ‘“ 𝑓:ω–ontoβ†’(𝐴 βŠ” 1o)))
Colors of variables: wff set class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708  DECID wdc 834   = wceq 1353  βˆƒwex 1492   ∈ wcel 2148  βˆ€wral 2455  βˆƒwrex 2456  βˆ…c0 3422  ifcif 3534  βˆͺ cuni 3809   ↦ cmpt 4064  Tr wtr 4101  Ord word 4362  suc csuc 4365  Ο‰com 4589  βŸΆwf 5212  β€“ontoβ†’wfo 5214  β€˜cfv 5216  1oc1o 6409   βŠ” cdju 7035  inlcinl 7043  inrcinr 7044
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 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-fal 1359  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-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-1st 6140  df-2nd 6141  df-1o 6416  df-dju 7036  df-inl 7045  df-inr 7046
This theorem is referenced by:  ctm  7107
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