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Theorem fodomr 8320
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 8168 . . . 4 Rel ≼
21brrelex2i 5331 . . 3 (𝐵𝐴𝐴 ∈ V)
32adantl 473 . 2 ((∅ ≺ 𝐵𝐵𝐴) → 𝐴 ∈ V)
41brrelex1i 5330 . . . 4 (𝐵𝐴𝐵 ∈ V)
5 0sdomg 8298 . . . . 5 (𝐵 ∈ V → (∅ ≺ 𝐵𝐵 ≠ ∅))
6 n0 4097 . . . . 5 (𝐵 ≠ ∅ ↔ ∃𝑧 𝑧𝐵)
75, 6syl6bb 278 . . . 4 (𝐵 ∈ V → (∅ ≺ 𝐵 ↔ ∃𝑧 𝑧𝐵))
84, 7syl 17 . . 3 (𝐵𝐴 → (∅ ≺ 𝐵 ↔ ∃𝑧 𝑧𝐵))
98biimpac 470 . 2 ((∅ ≺ 𝐵𝐵𝐴) → ∃𝑧 𝑧𝐵)
10 brdomi 8173 . . 3 (𝐵𝐴 → ∃𝑔 𝑔:𝐵1-1𝐴)
1110adantl 473 . 2 ((∅ ≺ 𝐵𝐵𝐴) → ∃𝑔 𝑔:𝐵1-1𝐴)
12 difexg 4971 . . . . . . . . . 10 (𝐴 ∈ V → (𝐴 ∖ ran 𝑔) ∈ V)
13 snex 5066 . . . . . . . . . 10 {𝑧} ∈ V
14 xpexg 7160 . . . . . . . . . 10 (((𝐴 ∖ ran 𝑔) ∈ V ∧ {𝑧} ∈ V) → ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ V)
1512, 13, 14sylancl 580 . . . . . . . . 9 (𝐴 ∈ V → ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ V)
16 vex 3353 . . . . . . . . . 10 𝑔 ∈ V
1716cnvex 7313 . . . . . . . . 9 𝑔 ∈ V
1815, 17jctil 515 . . . . . . . 8 (𝐴 ∈ V → (𝑔 ∈ V ∧ ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ V))
19 unexb 7158 . . . . . . . 8 ((𝑔 ∈ V ∧ ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ V) ↔ (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∈ V)
2018, 19sylib 209 . . . . . . 7 (𝐴 ∈ V → (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∈ V)
21 df-f1 6075 . . . . . . . . . . . . 13 (𝑔:𝐵1-1𝐴 ↔ (𝑔:𝐵𝐴 ∧ Fun 𝑔))
2221simprbi 490 . . . . . . . . . . . 12 (𝑔:𝐵1-1𝐴 → Fun 𝑔)
23 vex 3353 . . . . . . . . . . . . . 14 𝑧 ∈ V
2423fconst 6275 . . . . . . . . . . . . 13 ((𝐴 ∖ ran 𝑔) × {𝑧}):(𝐴 ∖ ran 𝑔)⟶{𝑧}
25 ffun 6228 . . . . . . . . . . . . 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 516 . . . . . . . . . . 11 (𝑔:𝐵1-1𝐴 → (Fun 𝑔 ∧ Fun ((𝐴 ∖ ran 𝑔) × {𝑧})))
28 df-rn 5290 . . . . . . . . . . . . . 14 ran 𝑔 = dom 𝑔
2928eqcomi 2774 . . . . . . . . . . . . 13 dom 𝑔 = ran 𝑔
3023snnz 4465 . . . . . . . . . . . . . 14 {𝑧} ≠ ∅
31 dmxp 5514 . . . . . . . . . . . . . 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 3976 . . . . . . . . . . . 12 (dom 𝑔 ∩ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∩ (𝐴 ∖ ran 𝑔))
34 disjdif 4202 . . . . . . . . . . . 12 (ran 𝑔 ∩ (𝐴 ∖ ran 𝑔)) = ∅
3533, 34eqtri 2787 . . . . . . . . . . 11 (dom 𝑔 ∩ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = ∅
36 funun 6115 . . . . . . . . . . 11 (((Fun 𝑔 ∧ Fun ((𝐴 ∖ ran 𝑔) × {𝑧})) ∧ (dom 𝑔 ∩ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = ∅) → Fun (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})))
3727, 35, 36sylancl 580 . . . . . . . . . 10 (𝑔:𝐵1-1𝐴 → Fun (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})))
3837adantl 473 . . . . . . . . 9 ((𝑧𝐵𝑔:𝐵1-1𝐴) → Fun (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})))
39 dmun 5501 . . . . . . . . . . . 12 dom (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (dom 𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧}))
4028uneq1i 3927 . . . . . . . . . . . 12 (ran 𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (dom 𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧}))
4132uneq2i 3928 . . . . . . . . . . . 12 (ran 𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔))
4239, 40, 413eqtr2i 2793 . . . . . . . . . . 11 dom (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔))
43 f1f 6285 . . . . . . . . . . . . 13 (𝑔:𝐵1-1𝐴𝑔:𝐵𝐴)
4443frnd 6232 . . . . . . . . . . . 12 (𝑔:𝐵1-1𝐴 → ran 𝑔𝐴)
45 undif 4211 . . . . . . . . . . . 12 (ran 𝑔𝐴 ↔ (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔)) = 𝐴)
4644, 45sylib 209 . . . . . . . . . . 11 (𝑔:𝐵1-1𝐴 → (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔)) = 𝐴)
4742, 46syl5eq 2811 . . . . . . . . . 10 (𝑔:𝐵1-1𝐴 → dom (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐴)
4847adantl 473 . . . . . . . . 9 ((𝑧𝐵𝑔:𝐵1-1𝐴) → dom (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐴)
49 df-fn 6073 . . . . . . . . 9 ((𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) Fn 𝐴 ↔ (Fun (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∧ dom (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐴))
5038, 48, 49sylanbrc 578 . . . . . . . 8 ((𝑧𝐵𝑔:𝐵1-1𝐴) → (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) Fn 𝐴)
51 rnun 5726 . . . . . . . . 9 ran (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧}))
52 dfdm4 5486 . . . . . . . . . . . 12 dom 𝑔 = ran 𝑔
53 f1dm 6289 . . . . . . . . . . . 12 (𝑔:𝐵1-1𝐴 → dom 𝑔 = 𝐵)
5452, 53syl5eqr 2813 . . . . . . . . . . 11 (𝑔:𝐵1-1𝐴 → ran 𝑔 = 𝐵)
5554uneq1d 3930 . . . . . . . . . 10 (𝑔:𝐵1-1𝐴 → (ran 𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = (𝐵 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})))
56 xpeq1 5293 . . . . . . . . . . . . . . . . 17 ((𝐴 ∖ ran 𝑔) = ∅ → ((𝐴 ∖ ran 𝑔) × {𝑧}) = (∅ × {𝑧}))
57 0xp 5371 . . . . . . . . . . . . . . . . 17 (∅ × {𝑧}) = ∅
5856, 57syl6eq 2815 . . . . . . . . . . . . . . . 16 ((𝐴 ∖ ran 𝑔) = ∅ → ((𝐴 ∖ ran 𝑔) × {𝑧}) = ∅)
5958rneqd 5523 . . . . . . . . . . . . . . 15 ((𝐴 ∖ ran 𝑔) = ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = ran ∅)
60 rn0 5548 . . . . . . . . . . . . . . 15 ran ∅ = ∅
6159, 60syl6eq 2815 . . . . . . . . . . . . . 14 ((𝐴 ∖ ran 𝑔) = ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = ∅)
62 0ss 4136 . . . . . . . . . . . . . 14 ∅ ⊆ 𝐵
6361, 62syl6eqss 3817 . . . . . . . . . . . . 13 ((𝐴 ∖ ran 𝑔) = ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵)
6463a1d 25 . . . . . . . . . . . 12 ((𝐴 ∖ ran 𝑔) = ∅ → (𝑧𝐵 → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵))
65 rnxp 5749 . . . . . . . . . . . . . . 15 ((𝐴 ∖ ran 𝑔) ≠ ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = {𝑧})
6665adantr 472 . . . . . . . . . . . . . 14 (((𝐴 ∖ ran 𝑔) ≠ ∅ ∧ 𝑧𝐵) → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = {𝑧})
67 snssi 4495 . . . . . . . . . . . . . . 15 (𝑧𝐵 → {𝑧} ⊆ 𝐵)
6867adantl 473 . . . . . . . . . . . . . 14 (((𝐴 ∖ ran 𝑔) ≠ ∅ ∧ 𝑧𝐵) → {𝑧} ⊆ 𝐵)
6966, 68eqsstrd 3801 . . . . . . . . . . . . 13 (((𝐴 ∖ ran 𝑔) ≠ ∅ ∧ 𝑧𝐵) → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵)
7069ex 401 . . . . . . . . . . . 12 ((𝐴 ∖ ran 𝑔) ≠ ∅ → (𝑧𝐵 → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵))
7164, 70pm2.61ine 3020 . . . . . . . . . . 11 (𝑧𝐵 → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵)
72 ssequn2 3950 . . . . . . . . . . 11 (ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵 ↔ (𝐵 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
7371, 72sylib 209 . . . . . . . . . 10 (𝑧𝐵 → (𝐵 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
7455, 73sylan9eqr 2821 . . . . . . . . 9 ((𝑧𝐵𝑔:𝐵1-1𝐴) → (ran 𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
7551, 74syl5eq 2811 . . . . . . . 8 ((𝑧𝐵𝑔:𝐵1-1𝐴) → ran (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)
76 df-fo 6076 . . . . . . . 8 ((𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴onto𝐵 ↔ ((𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) Fn 𝐴 ∧ ran (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵))
7750, 75, 76sylanbrc 578 . . . . . . 7 ((𝑧𝐵𝑔:𝐵1-1𝐴) → (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴onto𝐵)
78 foeq1 6296 . . . . . . . 8 (𝑓 = (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) → (𝑓:𝐴onto𝐵 ↔ (𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴onto𝐵))
7978spcegv 3447 . . . . . . 7 ((𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∈ V → ((𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴onto𝐵 → ∃𝑓 𝑓:𝐴onto𝐵))
8020, 77, 79syl2im 40 . . . . . 6 (𝐴 ∈ V → ((𝑧𝐵𝑔:𝐵1-1𝐴) → ∃𝑓 𝑓:𝐴onto𝐵))
8180expdimp 444 . . . . 5 ((𝐴 ∈ V ∧ 𝑧𝐵) → (𝑔:𝐵1-1𝐴 → ∃𝑓 𝑓:𝐴onto𝐵))
8281exlimdv 2028 . . . 4 ((𝐴 ∈ V ∧ 𝑧𝐵) → (∃𝑔 𝑔:𝐵1-1𝐴 → ∃𝑓 𝑓:𝐴onto𝐵))
8382ex 401 . . 3 (𝐴 ∈ V → (𝑧𝐵 → (∃𝑔 𝑔:𝐵1-1𝐴 → ∃𝑓 𝑓:𝐴onto𝐵)))
8483exlimdv 2028 . 2 (𝐴 ∈ V → (∃𝑧 𝑧𝐵 → (∃𝑔 𝑔:𝐵1-1𝐴 → ∃𝑓 𝑓:𝐴onto𝐵)))
853, 9, 11, 84syl3c 66 1 ((∅ ≺ 𝐵𝐵𝐴) → ∃𝑓 𝑓:𝐴onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wex 1874  wcel 2155  wne 2937  Vcvv 3350  cdif 3731  cun 3732  cin 3733  wss 3734  c0 4081  {csn 4336   class class class wbr 4811   × cxp 5277  ccnv 5278  dom cdm 5279  ran crn 5280  Fun wfun 6064   Fn wfn 6065  wf 6066  1-1wf1 6067  ontowfo 6068  cdom 8160  csdm 8161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-er 7949  df-en 8163  df-dom 8164  df-sdom 8165
This theorem is referenced by:  pwdom  8321  fodomfib  8449  domwdom  8688  iunfictbso  9190  fodomb  9603  brdom3  9605  konigthlem  9645  1stcfb  21531  ovoliunnul  23568  sigapildsys  30675  carsgclctunlem3  30832  ovoliunnfl  33878  voliunnfl  33880  volsupnfl  33881  nnfoctb  39867  caragenunicl  41381
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