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Theorem domssex2 8664
 Description: A corollary of disjenex 8662. 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 6550 . . . . 5 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
2 fex2 7623 . . . . 5 ((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
31, 2syl3an1 1160 . . . 4 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
4 f1stres 7698 . . . . 5 (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})):((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})⟶(𝐵 ∖ ran 𝐹)
5 difexg 5196 . . . . . . 7 (𝐵𝑊 → (𝐵 ∖ ran 𝐹) ∈ V)
653ad2ant3 1132 . . . . . 6 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐵 ∖ ran 𝐹) ∈ V)
7 snex 5298 . . . . . 6 {𝒫 ran 𝐴} ∈ V
8 xpexg 7456 . . . . . 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 7623 . . . . 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 1463 . . . 4 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) ∈ V)
12 unexg 7455 . . . 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 7614 . . 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 2798 . . . . . . 7 (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) = (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})))
1716domss2 8663 . . . . . 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 1139 . . . . 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 6590 . . . . 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 3939 . . . 4 ran (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ⊆ V
22 f1ss 6556 . . . 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 1141 . . 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 6545 . . 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 5693 . . . 4 (𝑔 = (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) → (𝑔𝐹) = ((𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ∘ 𝐹))
2827eqeq1d 2800 . . 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 633 . 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 3547 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 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Vcvv 3441   ∖ cdif 3878   ∪ cun 3879   ⊆ wss 3881  𝒫 cpw 4497  {csn 4525  ∪ cuni 4801   I cid 5425   × cxp 5518  ◡ccnv 5519  ran crn 5521   ↾ cres 5522   ∘ ccom 5524  ⟶wf 6321  –1-1→wf1 6322  –1-1-onto→wf1o 6324  1st c1st 7672 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-1st 7674  df-2nd 7675  df-en 8496 This theorem is referenced by: (None)
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