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Theorem f1finf1oOLD 9290
Description: Obsolete version of f1finf1o 9289 as of 4-Jan-2025. (Contributed by Jeff Madsen, 5-Jun-2010.) (Revised by Mario Carneiro, 27-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
f1finf1oOLD ((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) β†’ (𝐹:𝐴–1-1→𝐡 ↔ 𝐹:𝐴–1-1-onto→𝐡))
Proof of Theorem f1finf1oOLD
StepHypRef Expression
1 simpr 483 . . . 4 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ 𝐹:𝐴–1-1→𝐡)
2 f1f 6787 . . . . . . 7 (𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴⟢𝐡)
32adantl 480 . . . . . 6 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ 𝐹:𝐴⟢𝐡)
43ffnd 6718 . . . . 5 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ 𝐹 Fn 𝐴)
5 simpll 765 . . . . . 6 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ 𝐴 β‰ˆ 𝐡)
63frnd 6725 . . . . . . . . 9 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ ran 𝐹 βŠ† 𝐡)
7 df-pss 3961 . . . . . . . . . 10 (ran 𝐹 ⊊ 𝐡 ↔ (ran 𝐹 βŠ† 𝐡 ∧ ran 𝐹 β‰  𝐡))
87baib 534 . . . . . . . . 9 (ran 𝐹 βŠ† 𝐡 β†’ (ran 𝐹 ⊊ 𝐡 ↔ ran 𝐹 β‰  𝐡))
96, 8syl 17 . . . . . . . 8 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ (ran 𝐹 ⊊ 𝐡 ↔ ran 𝐹 β‰  𝐡))
10 simplr 767 . . . . . . . . . . . . 13 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ 𝐡 ∈ Fin)
11 relen 8962 . . . . . . . . . . . . . . 15 Rel β‰ˆ
1211brrelex1i 5729 . . . . . . . . . . . . . 14 (𝐴 β‰ˆ 𝐡 β†’ 𝐴 ∈ V)
135, 12syl 17 . . . . . . . . . . . . 13 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ 𝐴 ∈ V)
1410, 13elmapd 8852 . . . . . . . . . . . 12 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ (𝐹 ∈ (𝐡 ↑m 𝐴) ↔ 𝐹:𝐴⟢𝐡))
153, 14mpbird 256 . . . . . . . . . . 11 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ 𝐹 ∈ (𝐡 ↑m 𝐴))
16 f1f1orn 6843 . . . . . . . . . . . 12 (𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹)
1716adantl 480 . . . . . . . . . . 11 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹)
18 f1oen3g 8980 . . . . . . . . . . 11 ((𝐹 ∈ (𝐡 ↑m 𝐴) ∧ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹) β†’ 𝐴 β‰ˆ ran 𝐹)
1915, 17, 18syl2anc 582 . . . . . . . . . 10 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ 𝐴 β‰ˆ ran 𝐹)
20 php3 9230 . . . . . . . . . . . 12 ((𝐡 ∈ Fin ∧ ran 𝐹 ⊊ 𝐡) β†’ ran 𝐹 β‰Ί 𝐡)
2120ex 411 . . . . . . . . . . 11 (𝐡 ∈ Fin β†’ (ran 𝐹 ⊊ 𝐡 β†’ ran 𝐹 β‰Ί 𝐡))
2210, 21syl 17 . . . . . . . . . 10 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ (ran 𝐹 ⊊ 𝐡 β†’ ran 𝐹 β‰Ί 𝐡))
23 ensdomtr 9131 . . . . . . . . . 10 ((𝐴 β‰ˆ ran 𝐹 ∧ ran 𝐹 β‰Ί 𝐡) β†’ 𝐴 β‰Ί 𝐡)
2419, 22, 23syl6an 682 . . . . . . . . 9 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ (ran 𝐹 ⊊ 𝐡 β†’ 𝐴 β‰Ί 𝐡))
25 sdomnen 8995 . . . . . . . . 9 (𝐴 β‰Ί 𝐡 β†’ Β¬ 𝐴 β‰ˆ 𝐡)
2624, 25syl6 35 . . . . . . . 8 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ (ran 𝐹 ⊊ 𝐡 β†’ Β¬ 𝐴 β‰ˆ 𝐡))
279, 26sylbird 259 . . . . . . 7 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ (ran 𝐹 β‰  𝐡 β†’ Β¬ 𝐴 β‰ˆ 𝐡))
2827necon4ad 2949 . . . . . 6 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ (𝐴 β‰ˆ 𝐡 β†’ ran 𝐹 = 𝐡))
295, 28mpd 15 . . . . 5 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ ran 𝐹 = 𝐡)
30 df-fo 6549 . . . . 5 (𝐹:𝐴–onto→𝐡 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐡))
314, 29, 30sylanbrc 581 . . . 4 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ 𝐹:𝐴–onto→𝐡)
32 df-f1o 6550 . . . 4 (𝐹:𝐴–1-1-onto→𝐡 ↔ (𝐹:𝐴–1-1→𝐡 ∧ 𝐹:𝐴–onto→𝐡))
331, 31, 32sylanbrc 581 . . 3 (((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐡) β†’ 𝐹:𝐴–1-1-onto→𝐡)
3433ex 411 . 2 ((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) β†’ (𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴–1-1-onto→𝐡))
35 f1of1 6831 . 2 (𝐹:𝐴–1-1-onto→𝐡 β†’ 𝐹:𝐴–1-1→𝐡)
3634, 35impbid1 224 1 ((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) β†’ (𝐹:𝐴–1-1→𝐡 ↔ 𝐹:𝐴–1-1-onto→𝐡))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   β‰  wne 2930  Vcvv 3463   βŠ† wss 3941   ⊊ wpss 3942   class class class wbr 5144  ran crn 5674   Fn wfn 6538  βŸΆwf 6539  β€“1-1β†’wf1 6540  β€“ontoβ†’wfo 6541  β€“1-1-ontoβ†’wf1o 6542  (class class class)co 7413   ↑m cmap 8838   β‰ˆ cen 8954   β‰Ί csdm 8956  Fincfn 8957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-1o 8480  df-er 8718  df-map 8840  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961
This theorem is referenced by: (None)
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