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Theorem domssex2 8005
 Description: A corollary of disjenex 8003. 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 6014 . . . . 5 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
2 fex2 7014 . . . . 5 ((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
31, 2syl3an1 1351 . . . 4 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
4 f1stres 7081 . . . . . 6 (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})):((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})⟶(𝐵 ∖ ran 𝐹)
54a1i 11 . . . . 5 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})):((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})⟶(𝐵 ∖ ran 𝐹))
6 difexg 4735 . . . . . . 7 (𝐵𝑊 → (𝐵 ∖ ran 𝐹) ∈ V)
763ad2ant3 1077 . . . . . 6 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐵 ∖ ran 𝐹) ∈ V)
8 snex 4835 . . . . . 6 {𝒫 ran 𝐴} ∈ V
9 xpexg 6858 . . . . . 6 (((𝐵 ∖ ran 𝐹) ∈ V ∧ {𝒫 ran 𝐴} ∈ V) → ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}) ∈ V)
107, 8, 9sylancl 693 . . . . 5 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}) ∈ V)
11 fex2 7014 . . . . 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)
125, 10, 7, 11syl3anc 1318 . . . 4 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) ∈ V)
13 unexg 6857 . . . 4 ((𝐹 ∈ V ∧ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) ∈ V) → (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ∈ V)
143, 12, 13syl2anc 691 . . 3 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ∈ V)
15 cnvexg 7005 . . 3 ((𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ∈ V → (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ∈ V)
1614, 15syl 17 . 2 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ∈ V)
17 eqid 2610 . . . . . . 7 (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) = (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})))
1817domss2 8004 . . . . . 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)))
1918simp1d 1066 . . . . 5 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))):𝐵1-1-onto→ran (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))))
20 f1of1 6049 . . . . 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}))))
2119, 20syl 17 . . . 4 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))):𝐵1-1→ran (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))))
22 ssv 3588 . . . 4 ran (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ⊆ V
23 f1ss 6019 . . . 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)
2421, 22, 23sylancl 693 . . 3 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))):𝐵1-1→V)
2518simp3d 1068 . . 3 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → ((𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ∘ 𝐹) = ( I ↾ 𝐴))
2624, 25jca 553 . 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)))
27 f1eq1 6009 . . . 4 (𝑔 = (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) → (𝑔:𝐵1-1→V ↔ (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))):𝐵1-1→V))
28 coeq1 5201 . . . . 5 (𝑔 = (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) → (𝑔𝐹) = ((𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ∘ 𝐹))
2928eqeq1d 2612 . . . 4 (𝑔 = (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) → ((𝑔𝐹) = ( I ↾ 𝐴) ↔ ((𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ∘ 𝐹) = ( I ↾ 𝐴)))
3027, 29anbi12d 743 . . 3 (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))))
3130spcegv 3267 . 2 ((𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ∈ V → (((𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))):𝐵1-1→V ∧ ((𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ∘ 𝐹) = ( I ↾ 𝐴)) → ∃𝑔(𝑔:𝐵1-1→V ∧ (𝑔𝐹) = ( I ↾ 𝐴))))
3216, 26, 31sylc 63 1 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → ∃𝑔(𝑔:𝐵1-1→V ∧ (𝑔𝐹) = ( I ↾ 𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372   I cid 4948   × cxp 5036  ◡ccnv 5037  ran crn 5039   ↾ cres 5040   ∘ ccom 5042  ⟶wf 5800  –1-1→wf1 5801  –1-1-onto→wf1o 5803  1st c1st 7057 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-1st 7059  df-2nd 7060  df-en 7842 This theorem is referenced by: (None)
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