Theorem fodomfiOLD 9347

Description: Obsolete version of fodomfi 9327 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:𝐴–onto→𝐵) → 𝐵 ≼ 𝐴)

Proof of Theorem fodomfiOLD

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		foima 6800	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵)
2	1	adantl 481	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → (𝐹 “ 𝐴) = 𝐵)
3		imaeq2 6048	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐹 “ 𝑥) = (𝐹 “ ∅))
4		ima0 6069	. . . . . . 7 ⊢ (𝐹 “ ∅) = ∅
5	3, 4	eqtrdi 2787	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐹 “ 𝑥) = ∅)
6		id 22	. . . . . 6 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
7	5, 6	breq12d 5137	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐹 “ 𝑥) ≼ 𝑥 ↔ ∅ ≼ ∅))
8	7	imbi2d 340	. . . 4 ⊢ (𝑥 = ∅ → ((𝐹 Fn 𝐴 → (𝐹 “ 𝑥) ≼ 𝑥) ↔ (𝐹 Fn 𝐴 → ∅ ≼ ∅)))
9		imaeq2 6048	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐹 “ 𝑥) = (𝐹 “ 𝑦))
10		id 22	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
11	9, 10	breq12d 5137	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐹 “ 𝑥) ≼ 𝑥 ↔ (𝐹 “ 𝑦) ≼ 𝑦))
12	11	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐹 Fn 𝐴 → (𝐹 “ 𝑥) ≼ 𝑥) ↔ (𝐹 Fn 𝐴 → (𝐹 “ 𝑦) ≼ 𝑦)))
13		imaeq2 6048	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹 “ 𝑥) = (𝐹 “ (𝑦 ∪ {𝑧})))
14		id 22	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → 𝑥 = (𝑦 ∪ {𝑧}))
15	13, 14	breq12d 5137	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐹 “ 𝑥) ≼ 𝑥 ↔ (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧})))
16	15	imbi2d 340	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐹 Fn 𝐴 → (𝐹 “ 𝑥) ≼ 𝑥) ↔ (𝐹 Fn 𝐴 → (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧}))))
17		imaeq2 6048	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐹 “ 𝑥) = (𝐹 “ 𝐴))
18		id 22	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
19	17, 18	breq12d 5137	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝐹 “ 𝑥) ≼ 𝑥 ↔ (𝐹 “ 𝐴) ≼ 𝐴))
20	19	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝐹 Fn 𝐴 → (𝐹 “ 𝑥) ≼ 𝑥) ↔ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) ≼ 𝐴)))
21		0ex 5282	. . . . . 6 ⊢ ∅ ∈ V
22	21	0dom 9125	. . . . 5 ⊢ ∅ ≼ ∅
23	22	a1i 11	. . . 4 ⊢ (𝐹 Fn 𝐴 → ∅ ≼ ∅)
24		fnfun 6643	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
25	24	ad2antrl 728	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → Fun 𝐹)
26		funressn 7154	. . . . . . . . . . . . 13 ⊢ (Fun 𝐹 → (𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹‘𝑧)⟩})
27		rnss 5924	. . . . . . . . . . . . 13 ⊢ ((𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹‘𝑧)⟩} → ran (𝐹 ↾ {𝑧}) ⊆ ran {⟨𝑧, (𝐹‘𝑧)⟩})
28	25, 26, 27	3syl 18	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → ran (𝐹 ↾ {𝑧}) ⊆ ran {⟨𝑧, (𝐹‘𝑧)⟩})
29		df-ima 5672	. . . . . . . . . . . 12 ⊢ (𝐹 “ {𝑧}) = ran (𝐹 ↾ {𝑧})
30		vex 3468	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
31	30	rnsnop 6218	. . . . . . . . . . . . 13 ⊢ ran {⟨𝑧, (𝐹‘𝑧)⟩} = {(𝐹‘𝑧)}
32	31	eqcomi 2745	. . . . . . . . . . . 12 ⊢ {(𝐹‘𝑧)} = ran {⟨𝑧, (𝐹‘𝑧)⟩}
33	28, 29, 32	3sstr4g 4017	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → (𝐹 “ {𝑧}) ⊆ {(𝐹‘𝑧)})
34		snex 5411	. . . . . . . . . . 11 ⊢ {(𝐹‘𝑧)} ∈ V
35		ssexg 5298	. . . . . . . . . . 11 ⊢ (((𝐹 “ {𝑧}) ⊆ {(𝐹‘𝑧)} ∧ {(𝐹‘𝑧)} ∈ V) → (𝐹 “ {𝑧}) ∈ V)
36	33, 34, 35	sylancl 586	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → (𝐹 “ {𝑧}) ∈ V)
37		fvi 6960	. . . . . . . . . 10 ⊢ ((𝐹 “ {𝑧}) ∈ V → ( I ‘(𝐹 “ {𝑧})) = (𝐹 “ {𝑧}))
38	36, 37	syl 17	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → ( I ‘(𝐹 “ {𝑧})) = (𝐹 “ {𝑧}))
39	38	uneq2d 4148	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → ((𝐹 “ 𝑦) ∪ ( I ‘(𝐹 “ {𝑧}))) = ((𝐹 “ 𝑦) ∪ (𝐹 “ {𝑧})))
40		imaundi 6143	. . . . . . . 8 ⊢ (𝐹 “ (𝑦 ∪ {𝑧})) = ((𝐹 “ 𝑦) ∪ (𝐹 “ {𝑧}))
41	39, 40	eqtr4di 2789	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → ((𝐹 “ 𝑦) ∪ ( I ‘(𝐹 “ {𝑧}))) = (𝐹 “ (𝑦 ∪ {𝑧})))
42		simprr 772	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → (𝐹 “ 𝑦) ≼ 𝑦)
43		ssdomg 9019	. . . . . . . . . . 11 ⊢ ({(𝐹‘𝑧)} ∈ V → ((𝐹 “ {𝑧}) ⊆ {(𝐹‘𝑧)} → (𝐹 “ {𝑧}) ≼ {(𝐹‘𝑧)}))
44	34, 33, 43	mpsyl 68	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → (𝐹 “ {𝑧}) ≼ {(𝐹‘𝑧)})
45		fvex 6894	. . . . . . . . . . . 12 ⊢ (𝐹‘𝑧) ∈ V
46	45	ensn1 9040	. . . . . . . . . . 11 ⊢ {(𝐹‘𝑧)} ≈ 1o
47	30	ensn1 9040	. . . . . . . . . . 11 ⊢ {𝑧} ≈ 1o
48	46, 47	entr4i 9030	. . . . . . . . . 10 ⊢ {(𝐹‘𝑧)} ≈ {𝑧}
49		domentr 9032	. . . . . . . . . 10 ⊢ (((𝐹 “ {𝑧}) ≼ {(𝐹‘𝑧)} ∧ {(𝐹‘𝑧)} ≈ {𝑧}) → (𝐹 “ {𝑧}) ≼ {𝑧})
50	44, 48, 49	sylancl 586	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → (𝐹 “ {𝑧}) ≼ {𝑧})
51	38, 50	eqbrtrd 5146	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → ( I ‘(𝐹 “ {𝑧})) ≼ {𝑧})
52		simplr 768	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → ¬ 𝑧 ∈ 𝑦)
53		disjsn 4692	. . . . . . . . 9 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
54	52, 53	sylibr 234	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → (𝑦 ∩ {𝑧}) = ∅)
55		undom 9078	. . . . . . . 8 ⊢ ((((𝐹 “ 𝑦) ≼ 𝑦 ∧ ( I ‘(𝐹 “ {𝑧})) ≼ {𝑧}) ∧ (𝑦 ∩ {𝑧}) = ∅) → ((𝐹 “ 𝑦) ∪ ( I ‘(𝐹 “ {𝑧}))) ≼ (𝑦 ∪ {𝑧}))
56	42, 51, 54, 55	syl21anc 837	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → ((𝐹 “ 𝑦) ∪ ( I ‘(𝐹 “ {𝑧}))) ≼ (𝑦 ∪ {𝑧}))
57	41, 56	eqbrtrrd 5148	. . . . . 6 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝐹 Fn 𝐴 ∧ (𝐹 “ 𝑦) ≼ 𝑦)) → (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧}))
58	57	exp32 420	. . . . 5 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (𝐹 Fn 𝐴 → ((𝐹 “ 𝑦) ≼ 𝑦 → (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧}))))
59	58	a2d 29	. . . 4 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((𝐹 Fn 𝐴 → (𝐹 “ 𝑦) ≼ 𝑦) → (𝐹 Fn 𝐴 → (𝐹 “ (𝑦 ∪ {𝑧})) ≼ (𝑦 ∪ {𝑧}))))
60	8, 12, 16, 20, 23, 59	findcard2s 9184	. . 3 ⊢ (𝐴 ∈ Fin → (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) ≼ 𝐴))
61		fofn 6797	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
62	60, 61	impel 505	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → (𝐹 “ 𝐴) ≼ 𝐴)
63	2, 62	eqbrtrrd 5148	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ≼ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3464   ∪ cun 3929   ∩ cin 3930   ⊆ wss 3931  ∅c0 4313  {csn 4606  ⟨cop 4612   class class class wbr 5124   I cid 5552  ran crn 5660   ↾ cres 5661   “ cima 5662  Fun wfun 6530   Fn wfn 6531  –onto→wfo 6534  ‘cfv 6536  1_oc1o 8478   ≈ cen 8961   ≼ cdom 8962  Fincfn 8964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-om 7867  df-1o 8485  df-er 8724  df-en 8965  df-dom 8966  df-fin 8968

This theorem is referenced by: (None)