ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  exmidfodomrlemim GIF version

Theorem exmidfodomrlemim 7194
Description: Excluded middle implies the existence of a mapping from any set onto any inhabited set that it dominates. Proposition 1.1 of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 1-Jul-2022.)
Assertion
Ref Expression
exmidfodomrlemim (EXMID → ∀𝑥𝑦((∃𝑧 𝑧𝑦𝑦𝑥) → ∃𝑓 𝑓:𝑥onto𝑦))
Distinct variable groups:   𝑥,𝑓,𝑧   𝑦,𝑓,𝑧

Proof of Theorem exmidfodomrlemim
Dummy variables 𝑔 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brdomi 6743 . . . . 5 (𝑦𝑥 → ∃𝑔 𝑔:𝑦1-1𝑥)
21ad2antll 491 . . . 4 ((EXMID ∧ (∃𝑧 𝑧𝑦𝑦𝑥)) → ∃𝑔 𝑔:𝑦1-1𝑥)
3 df-f1 5217 . . . . . . . . . . . . . . . . 17 (𝑔:𝑦1-1𝑥 ↔ (𝑔:𝑦𝑥 ∧ Fun 𝑔))
43simprbi 275 . . . . . . . . . . . . . . . 16 (𝑔:𝑦1-1𝑥 → Fun 𝑔)
5 vex 2740 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
65fconst 5407 . . . . . . . . . . . . . . . . 17 ((𝑥 ∖ ran 𝑔) × {𝑧}):(𝑥 ∖ ran 𝑔)⟶{𝑧}
7 ffun 5364 . . . . . . . . . . . . . . . . 17 (((𝑥 ∖ ran 𝑔) × {𝑧}):(𝑥 ∖ ran 𝑔)⟶{𝑧} → Fun ((𝑥 ∖ ran 𝑔) × {𝑧}))
86, 7ax-mp 5 . . . . . . . . . . . . . . . 16 Fun ((𝑥 ∖ ran 𝑔) × {𝑧})
94, 8jctir 313 . . . . . . . . . . . . . . 15 (𝑔:𝑦1-1𝑥 → (Fun 𝑔 ∧ Fun ((𝑥 ∖ ran 𝑔) × {𝑧})))
10 df-rn 4634 . . . . . . . . . . . . . . . . . 18 ran 𝑔 = dom 𝑔
1110eqcomi 2181 . . . . . . . . . . . . . . . . 17 dom 𝑔 = ran 𝑔
125snm 3711 . . . . . . . . . . . . . . . . . 18 𝑤 𝑤 ∈ {𝑧}
13 dmxpm 4843 . . . . . . . . . . . . . . . . . 18 (∃𝑤 𝑤 ∈ {𝑧} → dom ((𝑥 ∖ ran 𝑔) × {𝑧}) = (𝑥 ∖ ran 𝑔))
1412, 13ax-mp 5 . . . . . . . . . . . . . . . . 17 dom ((𝑥 ∖ ran 𝑔) × {𝑧}) = (𝑥 ∖ ran 𝑔)
1511, 14ineq12i 3334 . . . . . . . . . . . . . . . 16 (dom 𝑔 ∩ dom ((𝑥 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∩ (𝑥 ∖ ran 𝑔))
16 disjdif 3495 . . . . . . . . . . . . . . . 16 (ran 𝑔 ∩ (𝑥 ∖ ran 𝑔)) = ∅
1715, 16eqtri 2198 . . . . . . . . . . . . . . 15 (dom 𝑔 ∩ dom ((𝑥 ∖ ran 𝑔) × {𝑧})) = ∅
18 funun 5256 . . . . . . . . . . . . . . 15 (((Fun 𝑔 ∧ Fun ((𝑥 ∖ ran 𝑔) × {𝑧})) ∧ (dom 𝑔 ∩ dom ((𝑥 ∖ ran 𝑔) × {𝑧})) = ∅) → Fun (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})))
199, 17, 18sylancl 413 . . . . . . . . . . . . . 14 (𝑔:𝑦1-1𝑥 → Fun (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})))
2019adantl 277 . . . . . . . . . . . . 13 (((EXMID𝑧𝑦) ∧ 𝑔:𝑦1-1𝑥) → Fun (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})))
21 dmun 4830 . . . . . . . . . . . . . . 15 dom (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = (dom 𝑔 ∪ dom ((𝑥 ∖ ran 𝑔) × {𝑧}))
2210uneq1i 3285 . . . . . . . . . . . . . . 15 (ran 𝑔 ∪ dom ((𝑥 ∖ ran 𝑔) × {𝑧})) = (dom 𝑔 ∪ dom ((𝑥 ∖ ran 𝑔) × {𝑧}))
2314uneq2i 3286 . . . . . . . . . . . . . . 15 (ran 𝑔 ∪ dom ((𝑥 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝑥 ∖ ran 𝑔))
2421, 22, 233eqtr2i 2204 . . . . . . . . . . . . . 14 dom (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝑥 ∖ ran 𝑔))
25 f1rn 5418 . . . . . . . . . . . . . . . 16 (𝑔:𝑦1-1𝑥 → ran 𝑔𝑥)
2625adantl 277 . . . . . . . . . . . . . . 15 (((EXMID𝑧𝑦) ∧ 𝑔:𝑦1-1𝑥) → ran 𝑔𝑥)
27 exmidexmid 4193 . . . . . . . . . . . . . . . . 17 (EXMIDDECID 𝑢 ∈ ran 𝑔)
2827ralrimivw 2551 . . . . . . . . . . . . . . . 16 (EXMID → ∀𝑢𝑥 DECID 𝑢 ∈ ran 𝑔)
2928ad2antrr 488 . . . . . . . . . . . . . . 15 (((EXMID𝑧𝑦) ∧ 𝑔:𝑦1-1𝑥) → ∀𝑢𝑥 DECID 𝑢 ∈ ran 𝑔)
30 undifdcss 6916 . . . . . . . . . . . . . . 15 (𝑥 = (ran 𝑔 ∪ (𝑥 ∖ ran 𝑔)) ↔ (ran 𝑔𝑥 ∧ ∀𝑢𝑥 DECID 𝑢 ∈ ran 𝑔))
3126, 29, 30sylanbrc 417 . . . . . . . . . . . . . 14 (((EXMID𝑧𝑦) ∧ 𝑔:𝑦1-1𝑥) → 𝑥 = (ran 𝑔 ∪ (𝑥 ∖ ran 𝑔)))
3224, 31eqtr4id 2229 . . . . . . . . . . . . 13 (((EXMID𝑧𝑦) ∧ 𝑔:𝑦1-1𝑥) → dom (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑥)
33 df-fn 5215 . . . . . . . . . . . . 13 ((𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) Fn 𝑥 ↔ (Fun (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) ∧ dom (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑥))
3420, 32, 33sylanbrc 417 . . . . . . . . . . . 12 (((EXMID𝑧𝑦) ∧ 𝑔:𝑦1-1𝑥) → (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) Fn 𝑥)
35 rnun 5033 . . . . . . . . . . . . 13 ran (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧}))
36 dfdm4 4815 . . . . . . . . . . . . . . . 16 dom 𝑔 = ran 𝑔
37 f1dm 5422 . . . . . . . . . . . . . . . 16 (𝑔:𝑦1-1𝑥 → dom 𝑔 = 𝑦)
3836, 37eqtr3id 2224 . . . . . . . . . . . . . . 15 (𝑔:𝑦1-1𝑥 → ran 𝑔 = 𝑦)
3938uneq1d 3288 . . . . . . . . . . . . . 14 (𝑔:𝑦1-1𝑥 → (ran 𝑔 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧})) = (𝑦 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧})))
40 exmidexmid 4193 . . . . . . . . . . . . . . . . . 18 (EXMIDDECID𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔))
41 exmiddc 836 . . . . . . . . . . . . . . . . . 18 (DECID𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) → (∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) ∨ ¬ ∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔)))
4240, 41syl 14 . . . . . . . . . . . . . . . . 17 (EXMID → (∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) ∨ ¬ ∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔)))
43 rnxpm 5054 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) = {𝑧})
4443adantr 276 . . . . . . . . . . . . . . . . . . . 20 ((∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) ∧ 𝑧𝑦) → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) = {𝑧})
45 snssi 3735 . . . . . . . . . . . . . . . . . . . . 21 (𝑧𝑦 → {𝑧} ⊆ 𝑦)
4645adantl 277 . . . . . . . . . . . . . . . . . . . 20 ((∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) ∧ 𝑧𝑦) → {𝑧} ⊆ 𝑦)
4744, 46eqsstrd 3191 . . . . . . . . . . . . . . . . . . 19 ((∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) ∧ 𝑧𝑦) → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) ⊆ 𝑦)
4847ex 115 . . . . . . . . . . . . . . . . . 18 (∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) → (𝑧𝑦 → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) ⊆ 𝑦))
49 notm0 3443 . . . . . . . . . . . . . . . . . . 19 (¬ ∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) ↔ (𝑥 ∖ ran 𝑔) = ∅)
50 xpeq1 4637 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∖ ran 𝑔) = ∅ → ((𝑥 ∖ ran 𝑔) × {𝑧}) = (∅ × {𝑧}))
51 0xp 4703 . . . . . . . . . . . . . . . . . . . . . . . 24 (∅ × {𝑧}) = ∅
5250, 51eqtrdi 2226 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 ∖ ran 𝑔) = ∅ → ((𝑥 ∖ ran 𝑔) × {𝑧}) = ∅)
5352rneqd 4852 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∖ ran 𝑔) = ∅ → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) = ran ∅)
54 rn0 4879 . . . . . . . . . . . . . . . . . . . . . 22 ran ∅ = ∅
5553, 54eqtrdi 2226 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∖ ran 𝑔) = ∅ → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) = ∅)
56 0ss 3461 . . . . . . . . . . . . . . . . . . . . 21 ∅ ⊆ 𝑦
5755, 56eqsstrdi 3207 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∖ ran 𝑔) = ∅ → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) ⊆ 𝑦)
5857a1d 22 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∖ ran 𝑔) = ∅ → (𝑧𝑦 → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) ⊆ 𝑦))
5949, 58sylbi 121 . . . . . . . . . . . . . . . . . 18 (¬ ∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) → (𝑧𝑦 → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) ⊆ 𝑦))
6048, 59jaoi 716 . . . . . . . . . . . . . . . . 17 ((∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) ∨ ¬ ∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔)) → (𝑧𝑦 → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) ⊆ 𝑦))
6142, 60syl 14 . . . . . . . . . . . . . . . 16 (EXMID → (𝑧𝑦 → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) ⊆ 𝑦))
6261imp 124 . . . . . . . . . . . . . . 15 ((EXMID𝑧𝑦) → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) ⊆ 𝑦)
63 ssequn2 3308 . . . . . . . . . . . . . . 15 (ran ((𝑥 ∖ ran 𝑔) × {𝑧}) ⊆ 𝑦 ↔ (𝑦 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑦)
6462, 63sylib 122 . . . . . . . . . . . . . 14 ((EXMID𝑧𝑦) → (𝑦 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑦)
6539, 64sylan9eqr 2232 . . . . . . . . . . . . 13 (((EXMID𝑧𝑦) ∧ 𝑔:𝑦1-1𝑥) → (ran 𝑔 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑦)
6635, 65eqtrid 2222 . . . . . . . . . . . 12 (((EXMID𝑧𝑦) ∧ 𝑔:𝑦1-1𝑥) → ran (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑦)
67 df-fo 5218 . . . . . . . . . . . 12 ((𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})):𝑥onto𝑦 ↔ ((𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) Fn 𝑥 ∧ ran (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑦))
6834, 66, 67sylanbrc 417 . . . . . . . . . . 11 (((EXMID𝑧𝑦) ∧ 𝑔:𝑦1-1𝑥) → (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})):𝑥onto𝑦)
69 vex 2740 . . . . . . . . . . . . . 14 𝑔 ∈ V
7069cnvex 5163 . . . . . . . . . . . . 13 𝑔 ∈ V
71 vex 2740 . . . . . . . . . . . . . . 15 𝑥 ∈ V
72 difexg 4141 . . . . . . . . . . . . . . 15 (𝑥 ∈ V → (𝑥 ∖ ran 𝑔) ∈ V)
7371, 72ax-mp 5 . . . . . . . . . . . . . 14 (𝑥 ∖ ran 𝑔) ∈ V
745snex 4182 . . . . . . . . . . . . . 14 {𝑧} ∈ V
7573, 74xpex 4738 . . . . . . . . . . . . 13 ((𝑥 ∖ ran 𝑔) × {𝑧}) ∈ V
7670, 75unex 4438 . . . . . . . . . . . 12 (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) ∈ V
77 foeq1 5430 . . . . . . . . . . . 12 (𝑓 = (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) → (𝑓:𝑥onto𝑦 ↔ (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})):𝑥onto𝑦))
7876, 77spcev 2832 . . . . . . . . . . 11 ((𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})):𝑥onto𝑦 → ∃𝑓 𝑓:𝑥onto𝑦)
7968, 78syl 14 . . . . . . . . . 10 (((EXMID𝑧𝑦) ∧ 𝑔:𝑦1-1𝑥) → ∃𝑓 𝑓:𝑥onto𝑦)
8079an32s 568 . . . . . . . . 9 (((EXMID𝑔:𝑦1-1𝑥) ∧ 𝑧𝑦) → ∃𝑓 𝑓:𝑥onto𝑦)
8180ex 115 . . . . . . . 8 ((EXMID𝑔:𝑦1-1𝑥) → (𝑧𝑦 → ∃𝑓 𝑓:𝑥onto𝑦))
8281exlimdv 1819 . . . . . . 7 ((EXMID𝑔:𝑦1-1𝑥) → (∃𝑧 𝑧𝑦 → ∃𝑓 𝑓:𝑥onto𝑦))
8382imp 124 . . . . . 6 (((EXMID𝑔:𝑦1-1𝑥) ∧ ∃𝑧 𝑧𝑦) → ∃𝑓 𝑓:𝑥onto𝑦)
8483an32s 568 . . . . 5 (((EXMID ∧ ∃𝑧 𝑧𝑦) ∧ 𝑔:𝑦1-1𝑥) → ∃𝑓 𝑓:𝑥onto𝑦)
8584adantlrr 483 . . . 4 (((EXMID ∧ (∃𝑧 𝑧𝑦𝑦𝑥)) ∧ 𝑔:𝑦1-1𝑥) → ∃𝑓 𝑓:𝑥onto𝑦)
862, 85exlimddv 1898 . . 3 ((EXMID ∧ (∃𝑧 𝑧𝑦𝑦𝑥)) → ∃𝑓 𝑓:𝑥onto𝑦)
8786ex 115 . 2 (EXMID → ((∃𝑧 𝑧𝑦𝑦𝑥) → ∃𝑓 𝑓:𝑥onto𝑦))
8887alrimivv 1875 1 (EXMID → ∀𝑥𝑦((∃𝑧 𝑧𝑦𝑦𝑥) → ∃𝑓 𝑓:𝑥onto𝑦))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wo 708  DECID wdc 834  wal 1351   = wceq 1353  wex 1492  wcel 2148  wral 2455  Vcvv 2737  cdif 3126  cun 3127  cin 3128  wss 3129  c0 3422  {csn 3591   class class class wbr 4000  EXMIDwem 4191   × cxp 4621  ccnv 4622  dom cdm 4623  ran crn 4624  Fun wfun 5206   Fn wfn 5207  wf 5208  1-1wf1 5209  ontowfo 5210  cdom 6733
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 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430
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-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-exmid 4192  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-dom 6736
This theorem is referenced by:  exmidfodomr  7197
  Copyright terms: Public domain W3C validator