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Theorem fodjum 7143
Description: Lemma for fodjuomni 7146 and fodjumkv 7157. A condition which shows that 𝐴 is inhabited. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.)
Hypotheses
Ref Expression
fodjuf.fo (πœ‘ β†’ 𝐹:𝑂–ontoβ†’(𝐴 βŠ” 𝐡))
fodjuf.p 𝑃 = (𝑦 ∈ 𝑂 ↦ if(βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘¦) = (inlβ€˜π‘§), βˆ…, 1o))
fodjum.z (πœ‘ β†’ βˆƒπ‘€ ∈ 𝑂 (π‘ƒβ€˜π‘€) = βˆ…)
Assertion
Ref Expression
fodjum (πœ‘ β†’ βˆƒπ‘₯ π‘₯ ∈ 𝐴)
Distinct variable groups:   πœ‘,𝑦,𝑧   𝑦,𝑂,𝑧   𝑧,𝐴   𝑧,𝐡   𝑧,𝐹   𝑀,𝐴,π‘₯,𝑧   𝑦,𝐴,𝑀   𝑦,𝐹   πœ‘,𝑀
Allowed substitution hints:   πœ‘(π‘₯)   𝐡(π‘₯,𝑦,𝑀)   𝑃(π‘₯,𝑦,𝑧,𝑀)   𝐹(π‘₯,𝑀)   𝑂(π‘₯,𝑀)

Proof of Theorem fodjum
StepHypRef Expression
1 fodjum.z . 2 (πœ‘ β†’ βˆƒπ‘€ ∈ 𝑂 (π‘ƒβ€˜π‘€) = βˆ…)
2 1n0 6432 . . . . . . . . 9 1o β‰  βˆ…
32nesymi 2393 . . . . . . . 8 Β¬ βˆ… = 1o
43intnan 929 . . . . . . 7 Β¬ (Β¬ βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘€) = (inlβ€˜π‘§) ∧ βˆ… = 1o)
54a1i 9 . . . . . 6 ((πœ‘ ∧ (𝑀 ∈ 𝑂 ∧ (π‘ƒβ€˜π‘€) = βˆ…)) β†’ Β¬ (Β¬ βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘€) = (inlβ€˜π‘§) ∧ βˆ… = 1o))
6 simprr 531 . . . . . . . 8 ((πœ‘ ∧ (𝑀 ∈ 𝑂 ∧ (π‘ƒβ€˜π‘€) = βˆ…)) β†’ (π‘ƒβ€˜π‘€) = βˆ…)
7 fodjuf.p . . . . . . . . 9 𝑃 = (𝑦 ∈ 𝑂 ↦ if(βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘¦) = (inlβ€˜π‘§), βˆ…, 1o))
8 fveqeq2 5524 . . . . . . . . . . 11 (𝑦 = 𝑀 β†’ ((πΉβ€˜π‘¦) = (inlβ€˜π‘§) ↔ (πΉβ€˜π‘€) = (inlβ€˜π‘§)))
98rexbidv 2478 . . . . . . . . . 10 (𝑦 = 𝑀 β†’ (βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘¦) = (inlβ€˜π‘§) ↔ βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘€) = (inlβ€˜π‘§)))
109ifbid 3555 . . . . . . . . 9 (𝑦 = 𝑀 β†’ if(βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘¦) = (inlβ€˜π‘§), βˆ…, 1o) = if(βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘€) = (inlβ€˜π‘§), βˆ…, 1o))
11 simprl 529 . . . . . . . . 9 ((πœ‘ ∧ (𝑀 ∈ 𝑂 ∧ (π‘ƒβ€˜π‘€) = βˆ…)) β†’ 𝑀 ∈ 𝑂)
12 peano1 4593 . . . . . . . . . . 11 βˆ… ∈ Ο‰
1312a1i 9 . . . . . . . . . 10 ((πœ‘ ∧ (𝑀 ∈ 𝑂 ∧ (π‘ƒβ€˜π‘€) = βˆ…)) β†’ βˆ… ∈ Ο‰)
14 1onn 6520 . . . . . . . . . . 11 1o ∈ Ο‰
1514a1i 9 . . . . . . . . . 10 ((πœ‘ ∧ (𝑀 ∈ 𝑂 ∧ (π‘ƒβ€˜π‘€) = βˆ…)) β†’ 1o ∈ Ο‰)
16 fodjuf.fo . . . . . . . . . . . 12 (πœ‘ β†’ 𝐹:𝑂–ontoβ†’(𝐴 βŠ” 𝐡))
1716fodjuomnilemdc 7141 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑀 ∈ 𝑂) β†’ DECID βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘€) = (inlβ€˜π‘§))
1817adantrr 479 . . . . . . . . . 10 ((πœ‘ ∧ (𝑀 ∈ 𝑂 ∧ (π‘ƒβ€˜π‘€) = βˆ…)) β†’ DECID βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘€) = (inlβ€˜π‘§))
1913, 15, 18ifcldcd 3570 . . . . . . . . 9 ((πœ‘ ∧ (𝑀 ∈ 𝑂 ∧ (π‘ƒβ€˜π‘€) = βˆ…)) β†’ if(βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘€) = (inlβ€˜π‘§), βˆ…, 1o) ∈ Ο‰)
207, 10, 11, 19fvmptd3 5609 . . . . . . . 8 ((πœ‘ ∧ (𝑀 ∈ 𝑂 ∧ (π‘ƒβ€˜π‘€) = βˆ…)) β†’ (π‘ƒβ€˜π‘€) = if(βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘€) = (inlβ€˜π‘§), βˆ…, 1o))
216, 20eqtr3d 2212 . . . . . . 7 ((πœ‘ ∧ (𝑀 ∈ 𝑂 ∧ (π‘ƒβ€˜π‘€) = βˆ…)) β†’ βˆ… = if(βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘€) = (inlβ€˜π‘§), βˆ…, 1o))
22 eqifdc 3569 . . . . . . . 8 (DECID βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘€) = (inlβ€˜π‘§) β†’ (βˆ… = if(βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘€) = (inlβ€˜π‘§), βˆ…, 1o) ↔ ((βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘€) = (inlβ€˜π‘§) ∧ βˆ… = βˆ…) ∨ (Β¬ βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘€) = (inlβ€˜π‘§) ∧ βˆ… = 1o))))
2318, 22syl 14 . . . . . . 7 ((πœ‘ ∧ (𝑀 ∈ 𝑂 ∧ (π‘ƒβ€˜π‘€) = βˆ…)) β†’ (βˆ… = if(βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘€) = (inlβ€˜π‘§), βˆ…, 1o) ↔ ((βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘€) = (inlβ€˜π‘§) ∧ βˆ… = βˆ…) ∨ (Β¬ βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘€) = (inlβ€˜π‘§) ∧ βˆ… = 1o))))
2421, 23mpbid 147 . . . . . 6 ((πœ‘ ∧ (𝑀 ∈ 𝑂 ∧ (π‘ƒβ€˜π‘€) = βˆ…)) β†’ ((βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘€) = (inlβ€˜π‘§) ∧ βˆ… = βˆ…) ∨ (Β¬ βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘€) = (inlβ€˜π‘§) ∧ βˆ… = 1o)))
255, 24ecased 1349 . . . . 5 ((πœ‘ ∧ (𝑀 ∈ 𝑂 ∧ (π‘ƒβ€˜π‘€) = βˆ…)) β†’ (βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘€) = (inlβ€˜π‘§) ∧ βˆ… = βˆ…))
2625simpld 112 . . . 4 ((πœ‘ ∧ (𝑀 ∈ 𝑂 ∧ (π‘ƒβ€˜π‘€) = βˆ…)) β†’ βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘€) = (inlβ€˜π‘§))
27 rexm 3522 . . . 4 (βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘€) = (inlβ€˜π‘§) β†’ βˆƒπ‘§ 𝑧 ∈ 𝐴)
2826, 27syl 14 . . 3 ((πœ‘ ∧ (𝑀 ∈ 𝑂 ∧ (π‘ƒβ€˜π‘€) = βˆ…)) β†’ βˆƒπ‘§ 𝑧 ∈ 𝐴)
29 eleq1w 2238 . . . 4 (𝑧 = π‘₯ β†’ (𝑧 ∈ 𝐴 ↔ π‘₯ ∈ 𝐴))
3029cbvexv 1918 . . 3 (βˆƒπ‘§ 𝑧 ∈ 𝐴 ↔ βˆƒπ‘₯ π‘₯ ∈ 𝐴)
3128, 30sylib 122 . 2 ((πœ‘ ∧ (𝑀 ∈ 𝑂 ∧ (π‘ƒβ€˜π‘€) = βˆ…)) β†’ βˆƒπ‘₯ π‘₯ ∈ 𝐴)
321, 31rexlimddv 2599 1 (πœ‘ β†’ βˆƒπ‘₯ π‘₯ ∈ 𝐴)
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  βˆƒwrex 2456  βˆ…c0 3422  ifcif 3534   ↦ cmpt 4064  Ο‰com 4589  β€“ontoβ†’wfo 5214  β€˜cfv 5216  1oc1o 6409   βŠ” cdju 7035  inlcinl 7043
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-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433
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-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-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-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:  fodjuomnilemres  7145  fodjumkvlemres  7156
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