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Theorem fodomr 7996
Description: There exists a mapping from a set onto any (nonempty) set that it dominates. (Contributed by NM, 23-Mar-2006.)
Assertion
Ref Expression
fodomr ((∅ ≺ 𝐵𝐵𝐴) → ∃𝑓 𝑓:𝐴onto𝐵)
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Proof of Theorem fodomr
Dummy variables 𝑔 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reldom 7847 . . . 4 Rel ≼
21brrelex2i 5083 . . 3 (𝐵𝐴𝐴 ∈ V)
32adantl 481 . 2 ((∅ ≺ 𝐵𝐵𝐴) → 𝐴 ∈ V)
41brrelexi 5082 . . . 4 (𝐵𝐴𝐵 ∈ V)
5 0sdomg 7974 . . . . 5 (𝐵 ∈ V → (∅ ≺ 𝐵𝐵 ≠ ∅))
6 n0 3890 . . . . 5 (𝐵 ≠ ∅ ↔ ∃𝑧 𝑧𝐵)
75, 6syl6bb 275 . . . 4 (𝐵 ∈ V → (∅ ≺ 𝐵 ↔ ∃𝑧 𝑧𝐵))
84, 7syl 17 . . 3 (𝐵𝐴 → (∅ ≺ 𝐵 ↔ ∃𝑧 𝑧𝐵))
98biimpac 502 . 2 ((∅ ≺ 𝐵𝐵𝐴) → ∃𝑧 𝑧𝐵)
10 brdomi 7852 . . 3 (𝐵𝐴 → ∃𝑔 𝑔:𝐵1-1𝐴)
1110adantl 481 . 2 ((∅ ≺ 𝐵𝐵𝐴) → ∃𝑔 𝑔:𝐵1-1𝐴)
12 difexg 4735 . . . . . . . . . 10 (𝐴 ∈ V → (𝐴 ∖ ran 𝑔) ∈ V)
13 snex 4835 . . . . . . . . . 10 {𝑧} ∈ V
14 xpexg 6858 . . . . . . . . . 10 (((𝐴 ∖ ran 𝑔) ∈ V ∧ {𝑧} ∈ V) → ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ V)
1512, 13, 14sylancl 693 . . . . . . . . 9 (𝐴 ∈ V → ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ V)
16 vex 3176 . . . . . . . . . 10 𝑔 ∈ V
1716cnvex 7006 . . . . . . . . 9 𝑔 ∈ V
1815, 17jctil 558 . . . . . . . 8 (𝐴 ∈ V → (𝑔 ∈ V ∧ ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ V))
19 unexb 6856 . . . . . . . 8 ((𝑔 ∈ V ∧ ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ V) ↔ (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∈ V)
2018, 19sylib 207 . . . . . . 7 (𝐴 ∈ V → (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∈ V)
21 df-f1 5809 . . . . . . . . . . . . 13 (𝑔:𝐵1-1𝐴 ↔ (𝑔:𝐵𝐴 ∧ Fun 𝑔))
2221simprbi 479 . . . . . . . . . . . 12 (𝑔:𝐵1-1𝐴 → Fun 𝑔)
23 vex 3176 . . . . . . . . . . . . . 14 𝑧 ∈ V
2423fconst 6004 . . . . . . . . . . . . 13 ((𝐴 ∖ ran 𝑔) × {𝑧}):(𝐴 ∖ ran 𝑔)⟶{𝑧}
25 ffun 5961 . . . . . . . . . . . . 13 (((𝐴 ∖ ran 𝑔) × {𝑧}):(𝐴 ∖ ran 𝑔)⟶{𝑧} → Fun ((𝐴 ∖ ran 𝑔) × {𝑧}))
2624, 25ax-mp 5 . . . . . . . . . . . 12 Fun ((𝐴 ∖ ran 𝑔) × {𝑧})
2722, 26jctir 559 . . . . . . . . . . 11 (𝑔:𝐵1-1𝐴 → (Fun 𝑔 ∧ Fun ((𝐴 ∖ ran 𝑔) × {𝑧})))
28 df-rn 5049 . . . . . . . . . . . . . 14 ran 𝑔 = dom 𝑔
2928eqcomi 2619 . . . . . . . . . . . . 13 dom 𝑔 = ran 𝑔
3023snnz 4252 . . . . . . . . . . . . . 14 {𝑧} ≠ ∅
31 dmxp 5265 . . . . . . . . . . . . . 14 ({𝑧} ≠ ∅ → dom ((𝐴 ∖ ran 𝑔) × {𝑧}) = (𝐴 ∖ ran 𝑔))
3230, 31ax-mp 5 . . . . . . . . . . . . 13 dom ((𝐴 ∖ ran 𝑔) × {𝑧}) = (𝐴 ∖ ran 𝑔)
3329, 32ineq12i 3774 . . . . . . . . . . . 12 (dom 𝑔 ∩ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∩ (𝐴 ∖ ran 𝑔))
34 disjdif 3992 . . . . . . . . . . . 12 (ran 𝑔 ∩ (𝐴 ∖ ran 𝑔)) = ∅
3533, 34eqtri 2632 . . . . . . . . . . 11 (dom 𝑔 ∩ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = ∅
36 funun 5846 . . . . . . . . . . 11 (((Fun 𝑔 ∧ Fun ((𝐴 ∖ ran 𝑔) × {𝑧})) ∧ (dom 𝑔 ∩ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = ∅) → Fun (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})))
3727, 35, 36sylancl 693 . . . . . . . . . 10 (𝑔:𝐵1-1𝐴 → Fun (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})))
3837adantl 481 . . . . . . . . 9 ((𝑧𝐵𝑔:𝐵1-1𝐴) → Fun (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})))
39 dmun 5253 . . . . . . . . . . . 12 dom (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (dom 𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧}))
4028uneq1i 3725 . . . . . . . . . . . 12 (ran 𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (dom 𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧}))
4132uneq2i 3726 . . . . . . . . . . . 12 (ran 𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔))
4239, 40, 413eqtr2i 2638 . . . . . . . . . . 11 dom (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔))
43 f1f 6014 . . . . . . . . . . . . 13 (𝑔:𝐵1-1𝐴𝑔:𝐵𝐴)
44 frn 5966 . . . . . . . . . . . . 13 (𝑔:𝐵𝐴 → ran 𝑔𝐴)
4543, 44syl 17 . . . . . . . . . . . 12 (𝑔:𝐵1-1𝐴 → ran 𝑔𝐴)
46 undif 4001 . . . . . . . . . . . 12 (ran 𝑔𝐴 ↔ (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔)) = 𝐴)
4745, 46sylib 207 . . . . . . . . . . 11 (𝑔:𝐵1-1𝐴 → (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔)) = 𝐴)
4842, 47syl5eq 2656 . . . . . . . . . 10 (𝑔:𝐵1-1𝐴 → dom (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐴)
4948adantl 481 . . . . . . . . 9 ((𝑧𝐵𝑔:𝐵1-1𝐴) → dom (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐴)
50 df-fn 5807 . . . . . . . . 9 ((𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) Fn 𝐴 ↔ (Fun (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∧ dom (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐴))
5138, 49, 50sylanbrc 695 . . . . . . . 8 ((𝑧𝐵𝑔:𝐵1-1𝐴) → (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) Fn 𝐴)
52 rnun 5460 . . . . . . . . 9 ran (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧}))
53 dfdm4 5238 . . . . . . . . . . . 12 dom 𝑔 = ran 𝑔
54 f1dm 6018 . . . . . . . . . . . 12 (𝑔:𝐵1-1𝐴 → dom 𝑔 = 𝐵)
5553, 54syl5eqr 2658 . . . . . . . . . . 11 (𝑔:𝐵1-1𝐴 → ran 𝑔 = 𝐵)
5655uneq1d 3728 . . . . . . . . . 10 (𝑔:𝐵1-1𝐴 → (ran 𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = (𝐵 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})))
57 xpeq1 5052 . . . . . . . . . . . . . . . . 17 ((𝐴 ∖ ran 𝑔) = ∅ → ((𝐴 ∖ ran 𝑔) × {𝑧}) = (∅ × {𝑧}))
58 0xp 5122 . . . . . . . . . . . . . . . . 17 (∅ × {𝑧}) = ∅
5957, 58syl6eq 2660 . . . . . . . . . . . . . . . 16 ((𝐴 ∖ ran 𝑔) = ∅ → ((𝐴 ∖ ran 𝑔) × {𝑧}) = ∅)
6059rneqd 5274 . . . . . . . . . . . . . . 15 ((𝐴 ∖ ran 𝑔) = ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = ran ∅)
61 rn0 5298 . . . . . . . . . . . . . . 15 ran ∅ = ∅
6260, 61syl6eq 2660 . . . . . . . . . . . . . 14 ((𝐴 ∖ ran 𝑔) = ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = ∅)
63 0ss 3924 . . . . . . . . . . . . . 14 ∅ ⊆ 𝐵
6462, 63syl6eqss 3618 . . . . . . . . . . . . 13 ((𝐴 ∖ ran 𝑔) = ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵)
6564a1d 25 . . . . . . . . . . . 12 ((𝐴 ∖ ran 𝑔) = ∅ → (𝑧𝐵 → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵))
66 rnxp 5483 . . . . . . . . . . . . . . 15 ((𝐴 ∖ ran 𝑔) ≠ ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = {𝑧})
6766adantr 480 . . . . . . . . . . . . . 14 (((𝐴 ∖ ran 𝑔) ≠ ∅ ∧ 𝑧𝐵) → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = {𝑧})
68 snssi 4280 . . . . . . . . . . . . . . 15 (𝑧𝐵 → {𝑧} ⊆ 𝐵)
6968adantl 481 . . . . . . . . . . . . . 14 (((𝐴 ∖ ran 𝑔) ≠ ∅ ∧ 𝑧𝐵) → {𝑧} ⊆ 𝐵)
7067, 69eqsstrd 3602 . . . . . . . . . . . . 13 (((𝐴 ∖ ran 𝑔) ≠ ∅ ∧ 𝑧𝐵) → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵)
7170ex 449 . . . . . . . . . . . 12 ((𝐴 ∖ ran 𝑔) ≠ ∅ → (𝑧𝐵 → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵))
7265, 71pm2.61ine 2865 . . . . . . . . . . 11 (𝑧𝐵 → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵)
73 ssequn2 3748 . . . . . . . . . . 11 (ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵 ↔ (𝐵 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
7472, 73sylib 207 . . . . . . . . . 10 (𝑧𝐵 → (𝐵 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
7556, 74sylan9eqr 2666 . . . . . . . . 9 ((𝑧𝐵𝑔:𝐵1-1𝐴) → (ran 𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
7652, 75syl5eq 2656 . . . . . . . 8 ((𝑧𝐵𝑔:𝐵1-1𝐴) → ran (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
77 df-fo 5810 . . . . . . . 8 ((𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴onto𝐵 ↔ ((𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) Fn 𝐴 ∧ ran (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵))
7851, 76, 77sylanbrc 695 . . . . . . 7 ((𝑧𝐵𝑔:𝐵1-1𝐴) → (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴onto𝐵)
79 foeq1 6024 . . . . . . . 8 (𝑓 = (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) → (𝑓:𝐴onto𝐵 ↔ (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴onto𝐵))
8079spcegv 3267 . . . . . . 7 ((𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∈ V → ((𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴onto𝐵 → ∃𝑓 𝑓:𝐴onto𝐵))
8120, 78, 80syl2im 39 . . . . . 6 (𝐴 ∈ V → ((𝑧𝐵𝑔:𝐵1-1𝐴) → ∃𝑓 𝑓:𝐴onto𝐵))
8281expdimp 452 . . . . 5 ((𝐴 ∈ V ∧ 𝑧𝐵) → (𝑔:𝐵1-1𝐴 → ∃𝑓 𝑓:𝐴onto𝐵))
8382exlimdv 1848 . . . 4 ((𝐴 ∈ V ∧ 𝑧𝐵) → (∃𝑔 𝑔:𝐵1-1𝐴 → ∃𝑓 𝑓:𝐴onto𝐵))
8483ex 449 . . 3 (𝐴 ∈ V → (𝑧𝐵 → (∃𝑔 𝑔:𝐵1-1𝐴 → ∃𝑓 𝑓:𝐴onto𝐵)))
8584exlimdv 1848 . 2 (𝐴 ∈ V → (∃𝑧 𝑧𝐵 → (∃𝑔 𝑔:𝐵1-1𝐴 → ∃𝑓 𝑓:𝐴onto𝐵)))
863, 9, 11, 85syl3c 64 1 ((∅ ≺ 𝐵𝐵𝐴) → ∃𝑓 𝑓:𝐴onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wex 1695  wcel 1977  wne 2780  Vcvv 3173  cdif 3537  cun 3538  cin 3539  wss 3540  c0 3874  {csn 4125   class class class wbr 4583   × cxp 5036  ccnv 5037  dom cdm 5038  ran crn 5039  Fun wfun 5798   Fn wfn 5799  wf 5800  1-1wf1 5801  ontowfo 5802  cdom 7839  csdm 7840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844
This theorem is referenced by:  pwdom  7997  fodomfib  8125  domwdom  8362  iunfictbso  8820  fodomb  9229  brdom3  9231  konigthlem  9269  1stcfb  21058  ovoliunnul  23082  sigapildsys  29552  carsgclctunlem3  29709  ovoliunnfl  32621  voliunnfl  32623  volsupnfl  32624  nnfoctb  38238  caragenunicl  39414
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