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Theorem fodomfi 8183
Description: An onto function implies dominance of domain over range, for finite sets. Unlike fodom 9288 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 6077 . . 3 (𝐹:𝐴onto𝐵 → (𝐹𝐴) = 𝐵)
21adantl 482 . 2 ((𝐴 ∈ Fin ∧ 𝐹:𝐴onto𝐵) → (𝐹𝐴) = 𝐵)
3 fofn 6074 . . . 4 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
4 imaeq2 5421 . . . . . . . 8 (𝑥 = ∅ → (𝐹𝑥) = (𝐹 “ ∅))
5 ima0 5440 . . . . . . . 8 (𝐹 “ ∅) = ∅
64, 5syl6eq 2671 . . . . . . 7 (𝑥 = ∅ → (𝐹𝑥) = ∅)
7 id 22 . . . . . . 7 (𝑥 = ∅ → 𝑥 = ∅)
86, 7breq12d 4626 . . . . . 6 (𝑥 = ∅ → ((𝐹𝑥) ≼ 𝑥 ↔ ∅ ≼ ∅))
98imbi2d 330 . . . . 5 (𝑥 = ∅ → ((𝐹 Fn 𝐴 → (𝐹𝑥) ≼ 𝑥) ↔ (𝐹 Fn 𝐴 → ∅ ≼ ∅)))
10 imaeq2 5421 . . . . . . 7 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
11 id 22 . . . . . . 7 (𝑥 = 𝑦𝑥 = 𝑦)
1210, 11breq12d 4626 . . . . . 6 (𝑥 = 𝑦 → ((𝐹𝑥) ≼ 𝑥 ↔ (𝐹𝑦) ≼ 𝑦))
1312imbi2d 330 . . . . 5 (𝑥 = 𝑦 → ((𝐹 Fn 𝐴 → (𝐹𝑥) ≼ 𝑥) ↔ (𝐹 Fn 𝐴 → (𝐹𝑦) ≼ 𝑦)))
14 imaeq2 5421 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹𝑥) = (𝐹 “ (𝑦 ∪ {𝑧})))
15 id 22 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → 𝑥 = (𝑦 ∪ {𝑧}))
1614, 15breq12d 4626 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐹𝑥) ≼ 𝑥 ↔ (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧})))
1716imbi2d 330 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐹 Fn 𝐴 → (𝐹𝑥) ≼ 𝑥) ↔ (𝐹 Fn 𝐴 → (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧}))))
18 imaeq2 5421 . . . . . . 7 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
19 id 22 . . . . . . 7 (𝑥 = 𝐴𝑥 = 𝐴)
2018, 19breq12d 4626 . . . . . 6 (𝑥 = 𝐴 → ((𝐹𝑥) ≼ 𝑥 ↔ (𝐹𝐴) ≼ 𝐴))
2120imbi2d 330 . . . . 5 (𝑥 = 𝐴 → ((𝐹 Fn 𝐴 → (𝐹𝑥) ≼ 𝑥) ↔ (𝐹 Fn 𝐴 → (𝐹𝐴) ≼ 𝐴)))
22 0ex 4750 . . . . . . 7 ∅ ∈ V
23220dom 8034 . . . . . 6 ∅ ≼ ∅
2423a1i 11 . . . . 5 (𝐹 Fn 𝐴 → ∅ ≼ ∅)
25 fnfun 5946 . . . . . . . . . . . . . . 15 (𝐹 Fn 𝐴 → Fun 𝐹)
2625ad2antrl 763 . . . . . . . . . . . . . 14 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → Fun 𝐹)
27 funressn 6380 . . . . . . . . . . . . . 14 (Fun 𝐹 → (𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹𝑧)⟩})
28 rnss 5314 . . . . . . . . . . . . . 14 ((𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹𝑧)⟩} → ran (𝐹 ↾ {𝑧}) ⊆ ran {⟨𝑧, (𝐹𝑧)⟩})
2926, 27, 283syl 18 . . . . . . . . . . . . 13 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → ran (𝐹 ↾ {𝑧}) ⊆ ran {⟨𝑧, (𝐹𝑧)⟩})
30 df-ima 5087 . . . . . . . . . . . . 13 (𝐹 “ {𝑧}) = ran (𝐹 ↾ {𝑧})
31 vex 3189 . . . . . . . . . . . . . . 15 𝑧 ∈ V
3231rnsnop 5575 . . . . . . . . . . . . . 14 ran {⟨𝑧, (𝐹𝑧)⟩} = {(𝐹𝑧)}
3332eqcomi 2630 . . . . . . . . . . . . 13 {(𝐹𝑧)} = ran {⟨𝑧, (𝐹𝑧)⟩}
3429, 30, 333sstr4g 3625 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → (𝐹 “ {𝑧}) ⊆ {(𝐹𝑧)})
35 snex 4869 . . . . . . . . . . . 12 {(𝐹𝑧)} ∈ V
36 ssexg 4764 . . . . . . . . . . . 12 (((𝐹 “ {𝑧}) ⊆ {(𝐹𝑧)} ∧ {(𝐹𝑧)} ∈ V) → (𝐹 “ {𝑧}) ∈ V)
3734, 35, 36sylancl 693 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → (𝐹 “ {𝑧}) ∈ V)
38 fvi 6212 . . . . . . . . . . 11 ((𝐹 “ {𝑧}) ∈ V → ( I ‘(𝐹 “ {𝑧})) = (𝐹 “ {𝑧}))
3937, 38syl 17 . . . . . . . . . 10 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → ( I ‘(𝐹 “ {𝑧})) = (𝐹 “ {𝑧}))
4039uneq2d 3745 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → ((𝐹𝑦) ∪ ( I ‘(𝐹 “ {𝑧}))) = ((𝐹𝑦) ∪ (𝐹 “ {𝑧})))
41 imaundi 5504 . . . . . . . . 9 (𝐹 “ (𝑦 ∪ {𝑧})) = ((𝐹𝑦) ∪ (𝐹 “ {𝑧}))
4240, 41syl6eqr 2673 . . . . . . . 8 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → ((𝐹𝑦) ∪ ( I ‘(𝐹 “ {𝑧}))) = (𝐹 “ (𝑦 ∪ {𝑧})))
43 simprr 795 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → (𝐹𝑦) ≼ 𝑦)
44 ssdomg 7945 . . . . . . . . . . . 12 ({(𝐹𝑧)} ∈ V → ((𝐹 “ {𝑧}) ⊆ {(𝐹𝑧)} → (𝐹 “ {𝑧}) ≼ {(𝐹𝑧)}))
4535, 34, 44mpsyl 68 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → (𝐹 “ {𝑧}) ≼ {(𝐹𝑧)})
46 fvex 6158 . . . . . . . . . . . . 13 (𝐹𝑧) ∈ V
4746ensn1 7964 . . . . . . . . . . . 12 {(𝐹𝑧)} ≈ 1𝑜
4831ensn1 7964 . . . . . . . . . . . 12 {𝑧} ≈ 1𝑜
4947, 48entr4i 7957 . . . . . . . . . . 11 {(𝐹𝑧)} ≈ {𝑧}
50 domentr 7959 . . . . . . . . . . 11 (((𝐹 “ {𝑧}) ≼ {(𝐹𝑧)} ∧ {(𝐹𝑧)} ≈ {𝑧}) → (𝐹 “ {𝑧}) ≼ {𝑧})
5145, 49, 50sylancl 693 . . . . . . . . . 10 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → (𝐹 “ {𝑧}) ≼ {𝑧})
5239, 51eqbrtrd 4635 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → ( I ‘(𝐹 “ {𝑧})) ≼ {𝑧})
53 simplr 791 . . . . . . . . . 10 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → ¬ 𝑧𝑦)
54 disjsn 4216 . . . . . . . . . 10 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑦)
5553, 54sylibr 224 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → (𝑦 ∩ {𝑧}) = ∅)
56 undom 7992 . . . . . . . . 9 ((((𝐹𝑦) ≼ 𝑦 ∧ ( I ‘(𝐹 “ {𝑧})) ≼ {𝑧}) ∧ (𝑦 ∩ {𝑧}) = ∅) → ((𝐹𝑦) ∪ ( I ‘(𝐹 “ {𝑧}))) ≼ (𝑦 ∪ {𝑧}))
5743, 52, 55, 56syl21anc 1322 . . . . . . . 8 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → ((𝐹𝑦) ∪ ( I ‘(𝐹 “ {𝑧}))) ≼ (𝑦 ∪ {𝑧}))
5842, 57eqbrtrrd 4637 . . . . . . 7 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧}))
5958exp32 630 . . . . . 6 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (𝐹 Fn 𝐴 → ((𝐹𝑦) ≼ 𝑦 → (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧}))))
6059a2d 29 . . . . 5 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → ((𝐹 Fn 𝐴 → (𝐹𝑦) ≼ 𝑦) → (𝐹 Fn 𝐴 → (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧}))))
619, 13, 17, 21, 24, 60findcard2s 8145 . . . 4 (𝐴 ∈ Fin → (𝐹 Fn 𝐴 → (𝐹𝐴) ≼ 𝐴))
623, 61syl5 34 . . 3 (𝐴 ∈ Fin → (𝐹:𝐴onto𝐵 → (𝐹𝐴) ≼ 𝐴))
6362imp 445 . 2 ((𝐴 ∈ Fin ∧ 𝐹:𝐴onto𝐵) → (𝐹𝐴) ≼ 𝐴)
642, 63eqbrtrrd 4637 1 ((𝐴 ∈ Fin ∧ 𝐹:𝐴onto𝐵) → 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  cun 3553  cin 3554  wss 3555  c0 3891  {csn 4148  cop 4154   class class class wbr 4613   I cid 4984  ran crn 5075  cres 5076  cima 5077  Fun wfun 5841   Fn wfn 5842  ontowfo 5845  cfv 5847  1𝑜c1o 7498  cen 7896  cdom 7897  Fincfn 7899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-om 7013  df-1o 7505  df-er 7687  df-en 7900  df-dom 7901  df-fin 7903
This theorem is referenced by:  fodomfib  8184  fofinf1o  8185  fidomdm  8187  fofi  8196  pwfilem  8204  cmpsub  21113  alexsubALT  21765  phpreu  33025  poimirlem26  33067
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