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Theorem domssex2 8524
 Description: A corollary of disjenex 8522. 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 6443 . . . . 5 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
2 fex2 7494 . . . . 5 ((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
31, 2syl3an1 1156 . . . 4 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
4 f1stres 7569 . . . . 5 (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})):((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})⟶(𝐵 ∖ ran 𝐹)
5 difexg 5122 . . . . . . 7 (𝐵𝑊 → (𝐵 ∖ ran 𝐹) ∈ V)
653ad2ant3 1128 . . . . . 6 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐵 ∖ ran 𝐹) ∈ V)
7 snex 5223 . . . . . 6 {𝒫 ran 𝐴} ∈ V
8 xpexg 7330 . . . . . 6 (((𝐵 ∖ ran 𝐹) ∈ V ∧ {𝒫 ran 𝐴} ∈ V) → ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}) ∈ V)
96, 7, 8sylancl 586 . . . . 5 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}) ∈ V)
10 fex2 7494 . . . . 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 1458 . . . 4 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) ∈ V)
12 unexg 7329 . . . 4 ((𝐹 ∈ V ∧ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})) ∈ V) → (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ∈ V)
133, 11, 12syl2anc 584 . . 3 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ∈ V)
14 cnvexg 7485 . . 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 2795 . . . . . . 7 (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) = (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})))
1716domss2 8523 . . . . . 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 1135 . . . . 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 6482 . . . . 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 3912 . . . 4 ran (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ⊆ V
22 f1ss 6448 . . . 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 586 . . 3 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))):𝐵1-1→V)
2417simp3d 1137 . . 3 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → ((𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ∘ 𝐹) = ( I ↾ 𝐴))
2523, 24jca 512 . 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 6438 . . 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 5614 . . . 4 (𝑔 = (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) → (𝑔𝐹) = ((𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴}))) ∘ 𝐹))
2827eqeq1d 2797 . . 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 630 . 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, 29elabd 3606 1 ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → ∃𝑔(𝑔:𝐵1-1→V ∧ (𝑔𝐹) = ( I ↾ 𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1080   = wceq 1522  ∃wex 1761   ∈ wcel 2081  Vcvv 3437   ∖ cdif 3856   ∪ cun 3857   ⊆ wss 3859  𝒫 cpw 4453  {csn 4472  ∪ cuni 4745   I cid 5347   × cxp 5441  ◡ccnv 5442  ran crn 5444   ↾ cres 5445   ∘ ccom 5447  ⟶wf 6221  –1-1→wf1 6222  –1-1-onto→wf1o 6224  1st c1st 7543 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-1st 7545  df-2nd 7546  df-en 8358 This theorem is referenced by: (None)
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