Theorem isinffi 9274

Description: An infinite set contains subsets equinumerous to every finite set. Extension of isinf 8584 from finite ordinals to all finite sets. (Contributed by Stefan O'Rear, 8-Oct-2014.)

Assertion

Ref	Expression
isinffi	⊢ ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐵–1-1→𝐴)

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Proof of Theorem isinffi

Dummy variables 𝑐 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ficardom 9243	. . 3 ⊢ (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
2		isinf 8584	. . 3 ⊢ (¬ 𝐴 ∈ Fin → ∀𝑎 ∈ ω ∃𝑐(𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ 𝑎))
3		breq2 4972	. . . . . 6 ⊢ (𝑎 = (card‘𝐵) → (𝑐 ≈ 𝑎 ↔ 𝑐 ≈ (card‘𝐵)))
4	3	anbi2d 628	. . . . 5 ⊢ (𝑎 = (card‘𝐵) → ((𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ 𝑎) ↔ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))))
5	4	exbidv 1903	. . . 4 ⊢ (𝑎 = (card‘𝐵) → (∃𝑐(𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ 𝑎) ↔ ∃𝑐(𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))))
6	5	rspcva 3559	. . 3 ⊢ (((card‘𝐵) ∈ ω ∧ ∀𝑎 ∈ ω ∃𝑐(𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ 𝑎)) → ∃𝑐(𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵)))
7	1, 2, 6	syl2anr 596	. 2 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ∃𝑐(𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵)))
8		simprr 769	. . . . . 6 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → 𝑐 ≈ (card‘𝐵))
9		ficardid 9244	. . . . . . 7 ⊢ (𝐵 ∈ Fin → (card‘𝐵) ≈ 𝐵)
10	9	ad2antlr 723	. . . . . 6 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → (card‘𝐵) ≈ 𝐵)
11		entr 8416	. . . . . 6 ⊢ ((𝑐 ≈ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝑐 ≈ 𝐵)
12	8, 10, 11	syl2anc 584	. . . . 5 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → 𝑐 ≈ 𝐵)
13	12	ensymd 8415	. . . 4 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → 𝐵 ≈ 𝑐)
14		bren 8373	. . . 4 ⊢ (𝐵 ≈ 𝑐 ↔ ∃𝑓 𝑓:𝐵–1-1-onto→𝑐)
15	13, 14	sylib 219	. . 3 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → ∃𝑓 𝑓:𝐵–1-1-onto→𝑐)
16		f1of1 6489	. . . . . 6 ⊢ (𝑓:𝐵–1-1-onto→𝑐 → 𝑓:𝐵–1-1→𝑐)
17		simplrl 773	. . . . . 6 ⊢ ((((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) ∧ 𝑓:𝐵–1-1-onto→𝑐) → 𝑐 ⊆ 𝐴)
18		f1ss 6455	. . . . . 6 ⊢ ((𝑓:𝐵–1-1→𝑐 ∧ 𝑐 ⊆ 𝐴) → 𝑓:𝐵–1-1→𝐴)
19	16, 17, 18	syl2an2 682	. . . . 5 ⊢ ((((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) ∧ 𝑓:𝐵–1-1-onto→𝑐) → 𝑓:𝐵–1-1→𝐴)
20	19	ex 413	. . . 4 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → (𝑓:𝐵–1-1-onto→𝑐 → 𝑓:𝐵–1-1→𝐴))
21	20	eximdv 1899	. . 3 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → (∃𝑓 𝑓:𝐵–1-1-onto→𝑐 → ∃𝑓 𝑓:𝐵–1-1→𝐴))
22	15, 21	mpd 15	. 2 ⊢ (((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝑐 ⊆ 𝐴 ∧ 𝑐 ≈ (card‘𝐵))) → ∃𝑓 𝑓:𝐵–1-1→𝐴)
23	7, 22	exlimddv 1917	1 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐵–1-1→𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1525  ∃wex 1765   ∈ wcel 2083  ∀wral 3107   ⊆ wss 3865   class class class wbr 4968  –1-1→wf1 6229  –1-1-onto→wf1o 6231  ‘cfv 6232  ωcom 7443   ≈ cen 8361  Fincfn 8364  cardccrd 9217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-om 7444  df-er 8146  df-en 8365  df-dom 8366  df-sdom 8367  df-fin 8368  df-card 9221

This theorem is referenced by:  fidomtri  9275  hashdom  13592  erdsze2lem1  32060  eldioph2lem2  38864