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Theorem fodomfi 9228
Description: An onto function implies dominance of domain over range, for finite sets. Unlike fodom 10418 for arbitrary sets, this theorem does not require the Axiom of Choice for its proof. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
fodomfi ((𝐴 ∈ Fin ∧ 𝐹:𝐴onto𝐵) → 𝐵𝐴)

Proof of Theorem fodomfi
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 foima 6759 . . 3 (𝐹:𝐴onto𝐵 → (𝐹𝐴) = 𝐵)
21adantl 483 . 2 ((𝐴 ∈ Fin ∧ 𝐹:𝐴onto𝐵) → (𝐹𝐴) = 𝐵)
3 imaeq2 6008 . . . . . . 7 (𝑥 = ∅ → (𝐹𝑥) = (𝐹 “ ∅))
4 ima0 6028 . . . . . . 7 (𝐹 “ ∅) = ∅
53, 4eqtrdi 2794 . . . . . 6 (𝑥 = ∅ → (𝐹𝑥) = ∅)
6 id 22 . . . . . 6 (𝑥 = ∅ → 𝑥 = ∅)
75, 6breq12d 5117 . . . . 5 (𝑥 = ∅ → ((𝐹𝑥) ≼ 𝑥 ↔ ∅ ≼ ∅))
87imbi2d 341 . . . 4 (𝑥 = ∅ → ((𝐹 Fn 𝐴 → (𝐹𝑥) ≼ 𝑥) ↔ (𝐹 Fn 𝐴 → ∅ ≼ ∅)))
9 imaeq2 6008 . . . . . 6 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
10 id 22 . . . . . 6 (𝑥 = 𝑦𝑥 = 𝑦)
119, 10breq12d 5117 . . . . 5 (𝑥 = 𝑦 → ((𝐹𝑥) ≼ 𝑥 ↔ (𝐹𝑦) ≼ 𝑦))
1211imbi2d 341 . . . 4 (𝑥 = 𝑦 → ((𝐹 Fn 𝐴 → (𝐹𝑥) ≼ 𝑥) ↔ (𝐹 Fn 𝐴 → (𝐹𝑦) ≼ 𝑦)))
13 imaeq2 6008 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹𝑥) = (𝐹 “ (𝑦 ∪ {𝑧})))
14 id 22 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → 𝑥 = (𝑦 ∪ {𝑧}))
1513, 14breq12d 5117 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐹𝑥) ≼ 𝑥 ↔ (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧})))
1615imbi2d 341 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐹 Fn 𝐴 → (𝐹𝑥) ≼ 𝑥) ↔ (𝐹 Fn 𝐴 → (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧}))))
17 imaeq2 6008 . . . . . 6 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
18 id 22 . . . . . 6 (𝑥 = 𝐴𝑥 = 𝐴)
1917, 18breq12d 5117 . . . . 5 (𝑥 = 𝐴 → ((𝐹𝑥) ≼ 𝑥 ↔ (𝐹𝐴) ≼ 𝐴))
2019imbi2d 341 . . . 4 (𝑥 = 𝐴 → ((𝐹 Fn 𝐴 → (𝐹𝑥) ≼ 𝑥) ↔ (𝐹 Fn 𝐴 → (𝐹𝐴) ≼ 𝐴)))
21 0ex 5263 . . . . . 6 ∅ ∈ V
22210dom 9009 . . . . 5 ∅ ≼ ∅
2322a1i 11 . . . 4 (𝐹 Fn 𝐴 → ∅ ≼ ∅)
24 fnfun 6600 . . . . . . . . . . . . . 14 (𝐹 Fn 𝐴 → Fun 𝐹)
2524ad2antrl 727 . . . . . . . . . . . . 13 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → Fun 𝐹)
26 funressn 7102 . . . . . . . . . . . . 13 (Fun 𝐹 → (𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹𝑧)⟩})
27 rnss 5893 . . . . . . . . . . . . 13 ((𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹𝑧)⟩} → ran (𝐹 ↾ {𝑧}) ⊆ ran {⟨𝑧, (𝐹𝑧)⟩})
2825, 26, 273syl 18 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → ran (𝐹 ↾ {𝑧}) ⊆ ran {⟨𝑧, (𝐹𝑧)⟩})
29 df-ima 5645 . . . . . . . . . . . 12 (𝐹 “ {𝑧}) = ran (𝐹 ↾ {𝑧})
30 vex 3448 . . . . . . . . . . . . . 14 𝑧 ∈ V
3130rnsnop 6175 . . . . . . . . . . . . 13 ran {⟨𝑧, (𝐹𝑧)⟩} = {(𝐹𝑧)}
3231eqcomi 2747 . . . . . . . . . . . 12 {(𝐹𝑧)} = ran {⟨𝑧, (𝐹𝑧)⟩}
3328, 29, 323sstr4g 3988 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → (𝐹 “ {𝑧}) ⊆ {(𝐹𝑧)})
34 snex 5387 . . . . . . . . . . 11 {(𝐹𝑧)} ∈ V
35 ssexg 5279 . . . . . . . . . . 11 (((𝐹 “ {𝑧}) ⊆ {(𝐹𝑧)} ∧ {(𝐹𝑧)} ∈ V) → (𝐹 “ {𝑧}) ∈ V)
3633, 34, 35sylancl 587 . . . . . . . . . 10 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → (𝐹 “ {𝑧}) ∈ V)
37 fvi 6915 . . . . . . . . . 10 ((𝐹 “ {𝑧}) ∈ V → ( I ‘(𝐹 “ {𝑧})) = (𝐹 “ {𝑧}))
3836, 37syl 17 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → ( I ‘(𝐹 “ {𝑧})) = (𝐹 “ {𝑧}))
3938uneq2d 4122 . . . . . . . 8 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → ((𝐹𝑦) ∪ ( I ‘(𝐹 “ {𝑧}))) = ((𝐹𝑦) ∪ (𝐹 “ {𝑧})))
40 imaundi 6101 . . . . . . . 8 (𝐹 “ (𝑦 ∪ {𝑧})) = ((𝐹𝑦) ∪ (𝐹 “ {𝑧}))
4139, 40eqtr4di 2796 . . . . . . 7 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → ((𝐹𝑦) ∪ ( I ‘(𝐹 “ {𝑧}))) = (𝐹 “ (𝑦 ∪ {𝑧})))
42 simprr 772 . . . . . . . 8 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → (𝐹𝑦) ≼ 𝑦)
43 ssdomg 8899 . . . . . . . . . . 11 ({(𝐹𝑧)} ∈ V → ((𝐹 “ {𝑧}) ⊆ {(𝐹𝑧)} → (𝐹 “ {𝑧}) ≼ {(𝐹𝑧)}))
4434, 33, 43mpsyl 68 . . . . . . . . . 10 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → (𝐹 “ {𝑧}) ≼ {(𝐹𝑧)})
45 fvex 6853 . . . . . . . . . . . 12 (𝐹𝑧) ∈ V
4645ensn1 8920 . . . . . . . . . . 11 {(𝐹𝑧)} ≈ 1o
4730ensn1 8920 . . . . . . . . . . 11 {𝑧} ≈ 1o
4846, 47entr4i 8910 . . . . . . . . . 10 {(𝐹𝑧)} ≈ {𝑧}
49 domentr 8912 . . . . . . . . . 10 (((𝐹 “ {𝑧}) ≼ {(𝐹𝑧)} ∧ {(𝐹𝑧)} ≈ {𝑧}) → (𝐹 “ {𝑧}) ≼ {𝑧})
5044, 48, 49sylancl 587 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → (𝐹 “ {𝑧}) ≼ {𝑧})
5138, 50eqbrtrd 5126 . . . . . . . 8 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → ( I ‘(𝐹 “ {𝑧})) ≼ {𝑧})
52 simplr 768 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → ¬ 𝑧𝑦)
53 disjsn 4671 . . . . . . . . 9 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑦)
5452, 53sylibr 233 . . . . . . . 8 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → (𝑦 ∩ {𝑧}) = ∅)
55 undom 8962 . . . . . . . 8 ((((𝐹𝑦) ≼ 𝑦 ∧ ( I ‘(𝐹 “ {𝑧})) ≼ {𝑧}) ∧ (𝑦 ∩ {𝑧}) = ∅) → ((𝐹𝑦) ∪ ( I ‘(𝐹 “ {𝑧}))) ≼ (𝑦 ∪ {𝑧}))
5642, 51, 54, 55syl21anc 837 . . . . . . 7 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → ((𝐹𝑦) ∪ ( I ‘(𝐹 “ {𝑧}))) ≼ (𝑦 ∪ {𝑧}))
5741, 56eqbrtrrd 5128 . . . . . 6 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧}))
5857exp32 422 . . . . 5 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (𝐹 Fn 𝐴 → ((𝐹𝑦) ≼ 𝑦 → (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧}))))
5958a2d 29 . . . 4 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → ((𝐹 Fn 𝐴 → (𝐹𝑦) ≼ 𝑦) → (𝐹 Fn 𝐴 → (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧}))))
608, 12, 16, 20, 23, 59findcard2s 9068 . . 3 (𝐴 ∈ Fin → (𝐹 Fn 𝐴 → (𝐹𝐴) ≼ 𝐴))
61 fofn 6756 . . 3 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
6260, 61impel 507 . 2 ((𝐴 ∈ Fin ∧ 𝐹:𝐴onto𝐵) → (𝐹𝐴) ≼ 𝐴)
632, 62eqbrtrrd 5128 1 ((𝐴 ∈ Fin ∧ 𝐹:𝐴onto𝐵) → 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3444  cun 3907  cin 3908  wss 3909  c0 4281  {csn 4585  cop 4591   class class class wbr 5104   I cid 5529  ran crn 5633  cres 5634  cima 5635  Fun wfun 6488   Fn wfn 6489  ontowfo 6492  cfv 6494  1oc1o 8398  cen 8839  cdom 8840  Fincfn 8842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7665
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-tr 5222  df-id 5530  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5587  df-we 5589  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-om 7796  df-1o 8405  df-er 8607  df-en 8843  df-dom 8844  df-fin 8846
This theorem is referenced by:  fodomfib  9229  fofinf1o  9230  fidomdm  9232  fofi  9241  pwfilemOLD  9249  cmpsub  22703  alexsubALT  23354  phpreu  35994  poimirlem26  36036
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