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Theorem exmidfodomrlemim 7025
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 6611 . . . . 5 (𝑦𝑥 → ∃𝑔 𝑔:𝑦1-1𝑥)
21ad2antll 482 . . . 4 ((EXMID ∧ (∃𝑧 𝑧𝑦𝑦𝑥)) → ∃𝑔 𝑔:𝑦1-1𝑥)
3 df-f1 5098 . . . . . . . . . . . . . . . . 17 (𝑔:𝑦1-1𝑥 ↔ (𝑔:𝑦𝑥 ∧ Fun 𝑔))
43simprbi 273 . . . . . . . . . . . . . . . 16 (𝑔:𝑦1-1𝑥 → Fun 𝑔)
5 vex 2663 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
65fconst 5288 . . . . . . . . . . . . . . . . 17 ((𝑥 ∖ ran 𝑔) × {𝑧}):(𝑥 ∖ ran 𝑔)⟶{𝑧}
7 ffun 5245 . . . . . . . . . . . . . . . . 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 311 . . . . . . . . . . . . . . 15 (𝑔:𝑦1-1𝑥 → (Fun 𝑔 ∧ Fun ((𝑥 ∖ ran 𝑔) × {𝑧})))
10 df-rn 4520 . . . . . . . . . . . . . . . . . 18 ran 𝑔 = dom 𝑔
1110eqcomi 2121 . . . . . . . . . . . . . . . . 17 dom 𝑔 = ran 𝑔
125snm 3613 . . . . . . . . . . . . . . . . . 18 𝑤 𝑤 ∈ {𝑧}
13 dmxpm 4729 . . . . . . . . . . . . . . . . . 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 3245 . . . . . . . . . . . . . . . 16 (dom 𝑔 ∩ dom ((𝑥 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∩ (𝑥 ∖ ran 𝑔))
16 disjdif 3405 . . . . . . . . . . . . . . . 16 (ran 𝑔 ∩ (𝑥 ∖ ran 𝑔)) = ∅
1715, 16eqtri 2138 . . . . . . . . . . . . . . 15 (dom 𝑔 ∩ dom ((𝑥 ∖ ran 𝑔) × {𝑧})) = ∅
18 funun 5137 . . . . . . . . . . . . . . 15 (((Fun 𝑔 ∧ Fun ((𝑥 ∖ ran 𝑔) × {𝑧})) ∧ (dom 𝑔 ∩ dom ((𝑥 ∖ ran 𝑔) × {𝑧})) = ∅) → Fun (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})))
199, 17, 18sylancl 409 . . . . . . . . . . . . . 14 (𝑔:𝑦1-1𝑥 → Fun (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})))
2019adantl 275 . . . . . . . . . . . . 13 (((EXMID𝑧𝑦) ∧ 𝑔:𝑦1-1𝑥) → Fun (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})))
21 f1rn 5299 . . . . . . . . . . . . . . . 16 (𝑔:𝑦1-1𝑥 → ran 𝑔𝑥)
2221adantl 275 . . . . . . . . . . . . . . 15 (((EXMID𝑧𝑦) ∧ 𝑔:𝑦1-1𝑥) → ran 𝑔𝑥)
23 exmidexmid 4090 . . . . . . . . . . . . . . . . 17 (EXMIDDECID 𝑢 ∈ ran 𝑔)
2423ralrimivw 2483 . . . . . . . . . . . . . . . 16 (EXMID → ∀𝑢𝑥 DECID 𝑢 ∈ ran 𝑔)
2524ad2antrr 479 . . . . . . . . . . . . . . 15 (((EXMID𝑧𝑦) ∧ 𝑔:𝑦1-1𝑥) → ∀𝑢𝑥 DECID 𝑢 ∈ ran 𝑔)
26 undifdcss 6779 . . . . . . . . . . . . . . 15 (𝑥 = (ran 𝑔 ∪ (𝑥 ∖ ran 𝑔)) ↔ (ran 𝑔𝑥 ∧ ∀𝑢𝑥 DECID 𝑢 ∈ ran 𝑔))
2722, 25, 26sylanbrc 413 . . . . . . . . . . . . . 14 (((EXMID𝑧𝑦) ∧ 𝑔:𝑦1-1𝑥) → 𝑥 = (ran 𝑔 ∪ (𝑥 ∖ ran 𝑔)))
28 dmun 4716 . . . . . . . . . . . . . . 15 dom (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = (dom 𝑔 ∪ dom ((𝑥 ∖ ran 𝑔) × {𝑧}))
2910uneq1i 3196 . . . . . . . . . . . . . . 15 (ran 𝑔 ∪ dom ((𝑥 ∖ ran 𝑔) × {𝑧})) = (dom 𝑔 ∪ dom ((𝑥 ∖ ran 𝑔) × {𝑧}))
3014uneq2i 3197 . . . . . . . . . . . . . . 15 (ran 𝑔 ∪ dom ((𝑥 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝑥 ∖ ran 𝑔))
3128, 29, 303eqtr2i 2144 . . . . . . . . . . . . . 14 dom (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝑥 ∖ ran 𝑔))
3227, 31syl6reqr 2169 . . . . . . . . . . . . 13 (((EXMID𝑧𝑦) ∧ 𝑔:𝑦1-1𝑥) → dom (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑥)
33 df-fn 5096 . . . . . . . . . . . . 13 ((𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) Fn 𝑥 ↔ (Fun (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) ∧ dom (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑥))
3420, 32, 33sylanbrc 413 . . . . . . . . . . . 12 (((EXMID𝑧𝑦) ∧ 𝑔:𝑦1-1𝑥) → (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) Fn 𝑥)
35 rnun 4917 . . . . . . . . . . . . 13 ran (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧}))
36 dfdm4 4701 . . . . . . . . . . . . . . . 16 dom 𝑔 = ran 𝑔
37 f1dm 5303 . . . . . . . . . . . . . . . 16 (𝑔:𝑦1-1𝑥 → dom 𝑔 = 𝑦)
3836, 37syl5eqr 2164 . . . . . . . . . . . . . . 15 (𝑔:𝑦1-1𝑥 → ran 𝑔 = 𝑦)
3938uneq1d 3199 . . . . . . . . . . . . . 14 (𝑔:𝑦1-1𝑥 → (ran 𝑔 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧})) = (𝑦 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧})))
40 exmidexmid 4090 . . . . . . . . . . . . . . . . . 18 (EXMIDDECID𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔))
41 exmiddc 806 . . . . . . . . . . . . . . . . . 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 4938 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) = {𝑧})
4443adantr 274 . . . . . . . . . . . . . . . . . . . 20 ((∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) ∧ 𝑧𝑦) → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) = {𝑧})
45 snssi 3634 . . . . . . . . . . . . . . . . . . . . 21 (𝑧𝑦 → {𝑧} ⊆ 𝑦)
4645adantl 275 . . . . . . . . . . . . . . . . . . . 20 ((∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) ∧ 𝑧𝑦) → {𝑧} ⊆ 𝑦)
4744, 46eqsstrd 3103 . . . . . . . . . . . . . . . . . . 19 ((∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) ∧ 𝑧𝑦) → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) ⊆ 𝑦)
4847ex 114 . . . . . . . . . . . . . . . . . 18 (∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) → (𝑧𝑦 → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) ⊆ 𝑦))
49 notm0 3353 . . . . . . . . . . . . . . . . . . 19 (¬ ∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) ↔ (𝑥 ∖ ran 𝑔) = ∅)
50 xpeq1 4523 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∖ ran 𝑔) = ∅ → ((𝑥 ∖ ran 𝑔) × {𝑧}) = (∅ × {𝑧}))
51 0xp 4589 . . . . . . . . . . . . . . . . . . . . . . . 24 (∅ × {𝑧}) = ∅
5250, 51syl6eq 2166 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 ∖ ran 𝑔) = ∅ → ((𝑥 ∖ ran 𝑔) × {𝑧}) = ∅)
5352rneqd 4738 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∖ ran 𝑔) = ∅ → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) = ran ∅)
54 rn0 4765 . . . . . . . . . . . . . . . . . . . . . 22 ran ∅ = ∅
5553, 54syl6eq 2166 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∖ ran 𝑔) = ∅ → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) = ∅)
56 0ss 3371 . . . . . . . . . . . . . . . . . . . . 21 ∅ ⊆ 𝑦
5755, 56eqsstrdi 3119 . . . . . . . . . . . . . . . . . . . 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 120 . . . . . . . . . . . . . . . . . 18 (¬ ∃𝑣 𝑣 ∈ (𝑥 ∖ ran 𝑔) → (𝑧𝑦 → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) ⊆ 𝑦))
6048, 59jaoi 690 . . . . . . . . . . . . . . . . 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 123 . . . . . . . . . . . . . . 15 ((EXMID𝑧𝑦) → ran ((𝑥 ∖ ran 𝑔) × {𝑧}) ⊆ 𝑦)
63 ssequn2 3219 . . . . . . . . . . . . . . 15 (ran ((𝑥 ∖ ran 𝑔) × {𝑧}) ⊆ 𝑦 ↔ (𝑦 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑦)
6462, 63sylib 121 . . . . . . . . . . . . . 14 ((EXMID𝑧𝑦) → (𝑦 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑦)
6539, 64sylan9eqr 2172 . . . . . . . . . . . . 13 (((EXMID𝑧𝑦) ∧ 𝑔:𝑦1-1𝑥) → (ran 𝑔 ∪ ran ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑦)
6635, 65syl5eq 2162 . . . . . . . . . . . 12 (((EXMID𝑧𝑦) ∧ 𝑔:𝑦1-1𝑥) → ran (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑦)
67 df-fo 5099 . . . . . . . . . . . 12 ((𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})):𝑥onto𝑦 ↔ ((𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) Fn 𝑥 ∧ ran (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) = 𝑦))
6834, 66, 67sylanbrc 413 . . . . . . . . . . 11 (((EXMID𝑧𝑦) ∧ 𝑔:𝑦1-1𝑥) → (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})):𝑥onto𝑦)
69 vex 2663 . . . . . . . . . . . . . 14 𝑔 ∈ V
7069cnvex 5047 . . . . . . . . . . . . 13 𝑔 ∈ V
71 vex 2663 . . . . . . . . . . . . . . 15 𝑥 ∈ V
72 difexg 4039 . . . . . . . . . . . . . . 15 (𝑥 ∈ V → (𝑥 ∖ ran 𝑔) ∈ V)
7371, 72ax-mp 5 . . . . . . . . . . . . . 14 (𝑥 ∖ ran 𝑔) ∈ V
745snex 4079 . . . . . . . . . . . . . 14 {𝑧} ∈ V
7573, 74xpex 4624 . . . . . . . . . . . . 13 ((𝑥 ∖ ran 𝑔) × {𝑧}) ∈ V
7670, 75unex 4332 . . . . . . . . . . . 12 (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) ∈ V
77 foeq1 5311 . . . . . . . . . . . 12 (𝑓 = (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})) → (𝑓:𝑥onto𝑦 ↔ (𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})):𝑥onto𝑦))
7876, 77spcev 2754 . . . . . . . . . . 11 ((𝑔 ∪ ((𝑥 ∖ ran 𝑔) × {𝑧})):𝑥onto𝑦 → ∃𝑓 𝑓:𝑥onto𝑦)
7968, 78syl 14 . . . . . . . . . 10 (((EXMID𝑧𝑦) ∧ 𝑔:𝑦1-1𝑥) → ∃𝑓 𝑓:𝑥onto𝑦)
8079an32s 542 . . . . . . . . 9 (((EXMID𝑔:𝑦1-1𝑥) ∧ 𝑧𝑦) → ∃𝑓 𝑓:𝑥onto𝑦)
8180ex 114 . . . . . . . 8 ((EXMID𝑔:𝑦1-1𝑥) → (𝑧𝑦 → ∃𝑓 𝑓:𝑥onto𝑦))
8281exlimdv 1775 . . . . . . 7 ((EXMID𝑔:𝑦1-1𝑥) → (∃𝑧 𝑧𝑦 → ∃𝑓 𝑓:𝑥onto𝑦))
8382imp 123 . . . . . 6 (((EXMID𝑔:𝑦1-1𝑥) ∧ ∃𝑧 𝑧𝑦) → ∃𝑓 𝑓:𝑥onto𝑦)
8483an32s 542 . . . . 5 (((EXMID ∧ ∃𝑧 𝑧𝑦) ∧ 𝑔:𝑦1-1𝑥) → ∃𝑓 𝑓:𝑥onto𝑦)
8584adantlrr 474 . . . 4 (((EXMID ∧ (∃𝑧 𝑧𝑦𝑦𝑥)) ∧ 𝑔:𝑦1-1𝑥) → ∃𝑓 𝑓:𝑥onto𝑦)
862, 85exlimddv 1854 . . 3 ((EXMID ∧ (∃𝑧 𝑧𝑦𝑦𝑥)) → ∃𝑓 𝑓:𝑥onto𝑦)
8786ex 114 . 2 (EXMID → ((∃𝑧 𝑧𝑦𝑦𝑥) → ∃𝑓 𝑓:𝑥onto𝑦))
8887alrimivv 1831 1 (EXMID → ∀𝑥𝑦((∃𝑧 𝑧𝑦𝑦𝑥) → ∃𝑓 𝑓:𝑥onto𝑦))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 682  DECID wdc 804  wal 1314   = wceq 1316  wex 1453  wcel 1465  wral 2393  Vcvv 2660  cdif 3038  cun 3039  cin 3040  wss 3041  c0 3333  {csn 3497   class class class wbr 3899  EXMIDwem 4088   × cxp 4507  ccnv 4508  dom cdm 4509  ran crn 4510  Fun wfun 5087   Fn wfn 5088  wf 5089  1-1wf1 5090  ontowfo 5091  cdom 6601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-exmid 4089  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-dom 6604
This theorem is referenced by:  exmidfodomr  7028
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