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Theorem fodomfiOLD 9371
Description: Obsolete version of fodomfi 9351 as of 20-Jun-2025. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Mario Carneiro, 16-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
fodomfiOLD ((𝐴 ∈ Fin ∧ 𝐹:𝐴onto𝐵) → 𝐵𝐴)

Proof of Theorem fodomfiOLD
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 foima 6824 . . 3 (𝐹:𝐴onto𝐵 → (𝐹𝐴) = 𝐵)
21adantl 481 . 2 ((𝐴 ∈ Fin ∧ 𝐹:𝐴onto𝐵) → (𝐹𝐴) = 𝐵)
3 imaeq2 6073 . . . . . . 7 (𝑥 = ∅ → (𝐹𝑥) = (𝐹 “ ∅))
4 ima0 6094 . . . . . . 7 (𝐹 “ ∅) = ∅
53, 4eqtrdi 2792 . . . . . 6 (𝑥 = ∅ → (𝐹𝑥) = ∅)
6 id 22 . . . . . 6 (𝑥 = ∅ → 𝑥 = ∅)
75, 6breq12d 5155 . . . . 5 (𝑥 = ∅ → ((𝐹𝑥) ≼ 𝑥 ↔ ∅ ≼ ∅))
87imbi2d 340 . . . 4 (𝑥 = ∅ → ((𝐹 Fn 𝐴 → (𝐹𝑥) ≼ 𝑥) ↔ (𝐹 Fn 𝐴 → ∅ ≼ ∅)))
9 imaeq2 6073 . . . . . 6 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
10 id 22 . . . . . 6 (𝑥 = 𝑦𝑥 = 𝑦)
119, 10breq12d 5155 . . . . 5 (𝑥 = 𝑦 → ((𝐹𝑥) ≼ 𝑥 ↔ (𝐹𝑦) ≼ 𝑦))
1211imbi2d 340 . . . 4 (𝑥 = 𝑦 → ((𝐹 Fn 𝐴 → (𝐹𝑥) ≼ 𝑥) ↔ (𝐹 Fn 𝐴 → (𝐹𝑦) ≼ 𝑦)))
13 imaeq2 6073 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹𝑥) = (𝐹 “ (𝑦 ∪ {𝑧})))
14 id 22 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → 𝑥 = (𝑦 ∪ {𝑧}))
1513, 14breq12d 5155 . . . . 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 6073 . . . . . 6 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
18 id 22 . . . . . 6 (𝑥 = 𝐴𝑥 = 𝐴)
1917, 18breq12d 5155 . . . . 5 (𝑥 = 𝐴 → ((𝐹𝑥) ≼ 𝑥 ↔ (𝐹𝐴) ≼ 𝐴))
2019imbi2d 340 . . . 4 (𝑥 = 𝐴 → ((𝐹 Fn 𝐴 → (𝐹𝑥) ≼ 𝑥) ↔ (𝐹 Fn 𝐴 → (𝐹𝐴) ≼ 𝐴)))
21 0ex 5306 . . . . . 6 ∅ ∈ V
22210dom 9147 . . . . 5 ∅ ≼ ∅
2322a1i 11 . . . 4 (𝐹 Fn 𝐴 → ∅ ≼ ∅)
24 fnfun 6667 . . . . . . . . . . . . . 14 (𝐹 Fn 𝐴 → Fun 𝐹)
2524ad2antrl 728 . . . . . . . . . . . . 13 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → Fun 𝐹)
26 funressn 7178 . . . . . . . . . . . . 13 (Fun 𝐹 → (𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹𝑧)⟩})
27 rnss 5949 . . . . . . . . . . . . 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 5697 . . . . . . . . . . . 12 (𝐹 “ {𝑧}) = ran (𝐹 ↾ {𝑧})
30 vex 3483 . . . . . . . . . . . . . 14 𝑧 ∈ V
3130rnsnop 6243 . . . . . . . . . . . . 13 ran {⟨𝑧, (𝐹𝑧)⟩} = {(𝐹𝑧)}
3231eqcomi 2745 . . . . . . . . . . . 12 {(𝐹𝑧)} = ran {⟨𝑧, (𝐹𝑧)⟩}
3328, 29, 323sstr4g 4036 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → (𝐹 “ {𝑧}) ⊆ {(𝐹𝑧)})
34 snex 5435 . . . . . . . . . . 11 {(𝐹𝑧)} ∈ V
35 ssexg 5322 . . . . . . . . . . 11 (((𝐹 “ {𝑧}) ⊆ {(𝐹𝑧)} ∧ {(𝐹𝑧)} ∈ V) → (𝐹 “ {𝑧}) ∈ V)
3633, 34, 35sylancl 586 . . . . . . . . . 10 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → (𝐹 “ {𝑧}) ∈ V)
37 fvi 6984 . . . . . . . . . 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 4167 . . . . . . . 8 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → ((𝐹𝑦) ∪ ( I ‘(𝐹 “ {𝑧}))) = ((𝐹𝑦) ∪ (𝐹 “ {𝑧})))
40 imaundi 6168 . . . . . . . 8 (𝐹 “ (𝑦 ∪ {𝑧})) = ((𝐹𝑦) ∪ (𝐹 “ {𝑧}))
4139, 40eqtr4di 2794 . . . . . . 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 9041 . . . . . . . . . . 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 6918 . . . . . . . . . . . 12 (𝐹𝑧) ∈ V
4645ensn1 9062 . . . . . . . . . . 11 {(𝐹𝑧)} ≈ 1o
4730ensn1 9062 . . . . . . . . . . 11 {𝑧} ≈ 1o
4846, 47entr4i 9052 . . . . . . . . . 10 {(𝐹𝑧)} ≈ {𝑧}
49 domentr 9054 . . . . . . . . . 10 (((𝐹 “ {𝑧}) ≼ {(𝐹𝑧)} ∧ {(𝐹𝑧)} ≈ {𝑧}) → (𝐹 “ {𝑧}) ≼ {𝑧})
5044, 48, 49sylancl 586 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → (𝐹 “ {𝑧}) ≼ {𝑧})
5138, 50eqbrtrd 5164 . . . . . . . 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 4710 . . . . . . . . 9 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑦)
5452, 53sylibr 234 . . . . . . . 8 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → (𝑦 ∩ {𝑧}) = ∅)
55 undom 9100 . . . . . . . 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 5166 . . . . . 6 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹𝑦) ≼ 𝑦)) → (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧}))
5857exp32 420 . . . . 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 9206 . . 3 (𝐴 ∈ Fin → (𝐹 Fn 𝐴 → (𝐹𝐴) ≼ 𝐴))
61 fofn 6821 . . 3 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
6260, 61impel 505 . 2 ((𝐴 ∈ Fin ∧ 𝐹:𝐴onto𝐵) → (𝐹𝐴) ≼ 𝐴)
632, 62eqbrtrrd 5166 1 ((𝐴 ∈ Fin ∧ 𝐹:𝐴onto𝐵) → 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1539  wcel 2107  Vcvv 3479  cun 3948  cin 3949  wss 3950  c0 4332  {csn 4625  cop 4631   class class class wbr 5142   I cid 5576  ran crn 5685  cres 5686  cima 5687  Fun wfun 6554   Fn wfn 6555  ontowfo 6558  cfv 6560  1oc1o 8500  cen 8983  cdom 8984  Fincfn 8986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-om 7889  df-1o 8507  df-er 8746  df-en 8987  df-dom 8988  df-fin 8990
This theorem is referenced by: (None)
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