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Theorem fodomfi 9378
Description: An onto function implies dominance of domain over range, for finite sets. Unlike fodomg 10591 for arbitrary sets, this theorem does not require the Axiom of Replacement nor the Axiom of Power Sets nor the Axiom of Choice for its proof. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Mario Carneiro, 16-Nov-2014.) Avoid ax-pow 5383. (Revised by BTernaryTau, 20-Jun-2025.)
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 6839 . . 3 (𝐹:𝐴onto𝐵 → (𝐹𝐴) = 𝐵)
21adantl 481 . 2 ((𝐴 ∈ Fin ∧ 𝐹:𝐴onto𝐵) → (𝐹𝐴) = 𝐵)
3 imaeq2 6085 . . . . . . 7 (𝑥 = ∅ → (𝐹𝑥) = (𝐹 “ ∅))
4 ima0 6106 . . . . . . 7 (𝐹 “ ∅) = ∅
53, 4eqtrdi 2796 . . . . . 6 (𝑥 = ∅ → (𝐹𝑥) = ∅)
6 id 22 . . . . . 6 (𝑥 = ∅ → 𝑥 = ∅)
75, 6breq12d 5179 . . . . 5 (𝑥 = ∅ → ((𝐹𝑥) ≼ 𝑥 ↔ ∅ ≼ ∅))
87imbi2d 340 . . . 4 (𝑥 = ∅ → ((𝐹 Fn 𝐴 → (𝐹𝑥) ≼ 𝑥) ↔ (𝐹 Fn 𝐴 → ∅ ≼ ∅)))
9 imaeq2 6085 . . . . . 6 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
10 id 22 . . . . . 6 (𝑥 = 𝑦𝑥 = 𝑦)
119, 10breq12d 5179 . . . . 5 (𝑥 = 𝑦 → ((𝐹𝑥) ≼ 𝑥 ↔ (𝐹𝑦) ≼ 𝑦))
1211imbi2d 340 . . . 4 (𝑥 = 𝑦 → ((𝐹 Fn 𝐴 → (𝐹𝑥) ≼ 𝑥) ↔ (𝐹 Fn 𝐴 → (𝐹𝑦) ≼ 𝑦)))
13 imaeq2 6085 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹𝑥) = (𝐹 “ (𝑦 ∪ {𝑧})))
14 id 22 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → 𝑥 = (𝑦 ∪ {𝑧}))
1513, 14breq12d 5179 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐹𝑥) ≼ 𝑥 ↔ (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧})))
1615imbi2d 340 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐹 Fn 𝐴 → (𝐹𝑥) ≼ 𝑥) ↔ (𝐹 Fn 𝐴 → (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧}))))
17 imaeq2 6085 . . . . . 6 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
18 id 22 . . . . . 6 (𝑥 = 𝐴𝑥 = 𝐴)
1917, 18breq12d 5179 . . . . 5 (𝑥 = 𝐴 → ((𝐹𝑥) ≼ 𝑥 ↔ (𝐹𝐴) ≼ 𝐴))
2019imbi2d 340 . . . 4 (𝑥 = 𝐴 → ((𝐹 Fn 𝐴 → (𝐹𝑥) ≼ 𝑥) ↔ (𝐹 Fn 𝐴 → (𝐹𝐴) ≼ 𝐴)))
21 0ex 5325 . . . . . 6 ∅ ∈ V
22210dom 9172 . . . . 5 ∅ ≼ ∅
2322a1i 11 . . . 4 (𝐹 Fn 𝐴 → ∅ ≼ ∅)
24 fnfun 6679 . . . . . . . . . . . . . 14 (𝐹 Fn 𝐴 → Fun 𝐹)
2524ad2antrl 727 . . . . . . . . . . . . 13 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → Fun 𝐹)
26 funressn 7193 . . . . . . . . . . . . 13 (Fun 𝐹 → (𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹𝑧)⟩})
27 rnss 5964 . . . . . . . . . . . . 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 5713 . . . . . . . . . . . 12 (𝐹 “ {𝑧}) = ran (𝐹 ↾ {𝑧})
30 vex 3492 . . . . . . . . . . . . . 14 𝑧 ∈ V
3130rnsnop 6255 . . . . . . . . . . . . 13 ran {⟨𝑧, (𝐹𝑧)⟩} = {(𝐹𝑧)}
3231eqcomi 2749 . . . . . . . . . . . 12 {(𝐹𝑧)} = ran {⟨𝑧, (𝐹𝑧)⟩}
3328, 29, 323sstr4g 4054 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → (𝐹 “ {𝑧}) ⊆ {(𝐹𝑧)})
34 snfi 9109 . . . . . . . . . . 11 {(𝐹𝑧)} ∈ Fin
35 ssexg 5341 . . . . . . . . . . 11 (((𝐹 “ {𝑧}) ⊆ {(𝐹𝑧)} ∧ {(𝐹𝑧)} ∈ Fin) → (𝐹 “ {𝑧}) ∈ V)
3633, 34, 35sylancl 585 . . . . . . . . . 10 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → (𝐹 “ {𝑧}) ∈ V)
37 fvi 6998 . . . . . . . . . 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 4191 . . . . . . . 8 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → ((𝐹𝑦) ∪ ( I ‘(𝐹 “ {𝑧}))) = ((𝐹𝑦) ∪ (𝐹 “ {𝑧})))
40 imaundi 6181 . . . . . . . 8 (𝐹 “ (𝑦 ∪ {𝑧})) = ((𝐹𝑦) ∪ (𝐹 “ {𝑧}))
4139, 40eqtr4di 2798 . . . . . . 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 ssdomfi 9262 . . . . . . . . . . 11 ({(𝐹𝑧)} ∈ Fin → ((𝐹 “ {𝑧}) ⊆ {(𝐹𝑧)} → (𝐹 “ {𝑧}) ≼ {(𝐹𝑧)}))
4434, 33, 43mpsyl 68 . . . . . . . . . 10 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → (𝐹 “ {𝑧}) ≼ {(𝐹𝑧)})
45 fvex 6933 . . . . . . . . . . 11 (𝐹𝑧) ∈ V
46 en2sn 9106 . . . . . . . . . . 11 (((𝐹𝑧) ∈ V ∧ 𝑧 ∈ V) → {(𝐹𝑧)} ≈ {𝑧})
4745, 30, 46mp2an 691 . . . . . . . . . 10 {(𝐹𝑧)} ≈ {𝑧}
48 endom 9039 . . . . . . . . . . 11 ({(𝐹𝑧)} ≈ {𝑧} → {(𝐹𝑧)} ≼ {𝑧})
49 domtrfi 9259 . . . . . . . . . . . 12 (({(𝐹𝑧)} ∈ Fin ∧ (𝐹 “ {𝑧}) ≼ {(𝐹𝑧)} ∧ {(𝐹𝑧)} ≼ {𝑧}) → (𝐹 “ {𝑧}) ≼ {𝑧})
5034, 49mp3an1 1448 . . . . . . . . . . 11 (((𝐹 “ {𝑧}) ≼ {(𝐹𝑧)} ∧ {(𝐹𝑧)} ≼ {𝑧}) → (𝐹 “ {𝑧}) ≼ {𝑧})
5148, 50sylan2 592 . . . . . . . . . 10 (((𝐹 “ {𝑧}) ≼ {(𝐹𝑧)} ∧ {(𝐹𝑧)} ≈ {𝑧}) → (𝐹 “ {𝑧}) ≼ {𝑧})
5244, 47, 51sylancl 585 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → (𝐹 “ {𝑧}) ≼ {𝑧})
5338, 52eqbrtrd 5188 . . . . . . . 8 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → ( I ‘(𝐹 “ {𝑧})) ≼ {𝑧})
54 simplr 768 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → ¬ 𝑧𝑦)
55 disjsn 4736 . . . . . . . . 9 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑦)
5654, 55sylibr 234 . . . . . . . 8 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → (𝑦 ∩ {𝑧}) = ∅)
57 undom 9125 . . . . . . . 8 ((((𝐹𝑦) ≼ 𝑦 ∧ ( I ‘(𝐹 “ {𝑧})) ≼ {𝑧}) ∧ (𝑦 ∩ {𝑧}) = ∅) → ((𝐹𝑦) ∪ ( I ‘(𝐹 “ {𝑧}))) ≼ (𝑦 ∪ {𝑧}))
5842, 53, 56, 57syl21anc 837 . . . . . . 7 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → ((𝐹𝑦) ∪ ( I ‘(𝐹 “ {𝑧}))) ≼ (𝑦 ∪ {𝑧}))
5941, 58eqbrtrrd 5190 . . . . . 6 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧}))
6059exp32 420 . . . . 5 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (𝐹 Fn 𝐴 → ((𝐹𝑦) ≼ 𝑦 → (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧}))))
6160a2d 29 . . . 4 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → ((𝐹 Fn 𝐴 → (𝐹𝑦) ≼ 𝑦) → (𝐹 Fn 𝐴 → (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧}))))
628, 12, 16, 20, 23, 61findcard2s 9231 . . 3 (𝐴 ∈ Fin → (𝐹 Fn 𝐴 → (𝐹𝐴) ≼ 𝐴))
63 fofn 6836 . . 3 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
6462, 63impel 505 . 2 ((𝐴 ∈ Fin ∧ 𝐹:𝐴onto𝐵) → (𝐹𝐴) ≼ 𝐴)
652, 64eqbrtrrd 5190 1 ((𝐴 ∈ Fin ∧ 𝐹:𝐴onto𝐵) → 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2108  Vcvv 3488  cun 3974  cin 3975  wss 3976  c0 4352  {csn 4648  cop 4654   class class class wbr 5166   I cid 5592  ran crn 5701  cres 5702  cima 5703  Fun wfun 6567   Fn wfn 6568  ontowfo 6571  cfv 6573  cen 9000  cdom 9001  Fincfn 9003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-om 7904  df-1o 8522  df-en 9004  df-dom 9005  df-fin 9007
This theorem is referenced by:  fofi  9379  fodomfib  9397  fodomfibOLD  9399  fofinf1o  9400  fidomdm  9402  pwfilemOLD  9416  cmpsub  23429  alexsubALT  24080  phpreu  37564  poimirlem26  37606  imadomfi  41959
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