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Theorem domssex2 8806
Description: A corollary of disjenex 8804. If 𝐹 is an injection from 𝐴 to 𝐵 then there is a right inverse 𝑔 of 𝐹 from 𝐵 to a superset of 𝐴. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
domssex2 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → ∃𝑔(𝑔:𝐵1-1→V ∧ (𝑔𝐹) = ( I ↾ 𝐴)))
Distinct variable groups:   𝐴,𝑔   𝐵,𝑔   𝑔,𝐹
Allowed substitution hints:   𝑉(𝑔)   𝑊(𝑔)

Proof of Theorem domssex2
StepHypRef Expression
1 f1f 6615 . . . . 5 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
2 fex2 7711 . . . . 5 ((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
31, 2syl3an1 1165 . . . 4 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
4 f1stres 7785 . . . . 5 (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})):((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})⟶(𝐵 ∖ ran 𝐹)
5 difexg 5220 . . . . . . 7 (𝐵𝑊 → (𝐵 ∖ ran 𝐹) ∈ V)
653ad2ant3 1137 . . . . . 6 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐵 ∖ ran 𝐹) ∈ V)
7 snex 5324 . . . . . 6 {𝒫 ran 𝐴} ∈ V
8 xpexg 7535 . . . . . 6 (((𝐵 ∖ ran 𝐹) ∈ V ∧ {𝒫 ran 𝐴} ∈ V) → ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}) ∈ V)
96, 7, 8sylancl 589 . . . . 5 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}) ∈ V)
10 fex2 7711 . . . . 5 (((1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})):((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})⟶(𝐵 ∖ ran 𝐹) ∧ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}) ∈ V ∧ (𝐵 ∖ ran 𝐹) ∈ V) → (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) ∈ V)
114, 9, 6, 10mp3an2i 1468 . . . 4 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) ∈ V)
12 unexg 7534 . . . 4 ((𝐹 ∈ V ∧ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) ∈ V) → (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ∈ V)
133, 11, 12syl2anc 587 . . 3 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ∈ V)
14 cnvexg 7702 . . 3 ((𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ∈ V → (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ∈ V)
1513, 14syl 17 . 2 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ∈ V)
16 eqid 2737 . . . . . . 7 (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) = (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})))
1716domss2 8805 . . . . . 6 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → ((𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))):𝐵1-1-onto→ran (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ∧ 𝐴 ⊆ ran (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ∧ ((𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ∘ 𝐹) = ( I ↾ 𝐴)))
1817simp1d 1144 . . . . 5 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))):𝐵1-1-onto→ran (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))))
19 f1of1 6660 . . . . 5 ((𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))):𝐵1-1-onto→ran (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) → (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))):𝐵1-1→ran (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))))
2018, 19syl 17 . . . 4 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))):𝐵1-1→ran (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))))
21 ssv 3925 . . . 4 ran (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ⊆ V
22 f1ss 6621 . . . 4 (((𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))):𝐵1-1→ran (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ∧ ran (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ⊆ V) → (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))):𝐵1-1→V)
2320, 21, 22sylancl 589 . . 3 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))):𝐵1-1→V)
2417simp3d 1146 . . 3 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → ((𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ∘ 𝐹) = ( I ↾ 𝐴))
2523, 24jca 515 . 2 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → ((𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))):𝐵1-1→V ∧ ((𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ∘ 𝐹) = ( I ↾ 𝐴)))
26 f1eq1 6610 . . 3 (𝑔 = (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) → (𝑔:𝐵1-1→V ↔ (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))):𝐵1-1→V))
27 coeq1 5726 . . . 4 (𝑔 = (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) → (𝑔𝐹) = ((𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ∘ 𝐹))
2827eqeq1d 2739 . . 3 (𝑔 = (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) → ((𝑔𝐹) = ( I ↾ 𝐴) ↔ ((𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ∘ 𝐹) = ( I ↾ 𝐴)))
2926, 28anbi12d 634 . 2 (𝑔 = (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) → ((𝑔:𝐵1-1→V ∧ (𝑔𝐹) = ( I ↾ 𝐴)) ↔ ((𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))):𝐵1-1→V ∧ ((𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ∘ 𝐹) = ( I ↾ 𝐴))))
3015, 25, 29spcedv 3513 1 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → ∃𝑔(𝑔:𝐵1-1→V ∧ (𝑔𝐹) = ( I ↾ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089   = wceq 1543  wex 1787  wcel 2110  Vcvv 3408  cdif 3863  cun 3864  wss 3866  𝒫 cpw 4513  {csn 4541   cuni 4819   I cid 5454   × cxp 5549  ccnv 5550  ran crn 5552  cres 5553  ccom 5555  wf 6376  1-1wf1 6377  1-1-ontowf1o 6379  1st c1st 7759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-1st 7761  df-2nd 7762  df-en 8627
This theorem is referenced by: (None)
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