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Theorem domssex2 9133
Description: A corollary of disjenex 9131. If 𝐹 is an injection from 𝐴 to 𝐡 then there is a right inverse 𝑔 of 𝐹 from 𝐡 to a superset of 𝐴. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
domssex2 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ βˆƒπ‘”(𝑔:𝐡–1-1β†’V ∧ (𝑔 ∘ 𝐹) = ( I β†Ύ 𝐴)))
Distinct variable groups:   𝐴,𝑔   𝐡,𝑔   𝑔,𝐹
Allowed substitution hints:   𝑉(𝑔)   π‘Š(𝑔)
Proof of Theorem domssex2
StepHypRef Expression
1 f1f 6784 . . . . 5 (𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴⟢𝐡)
2 fex2 7920 . . . . 5 ((𝐹:𝐴⟢𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ 𝐹 ∈ V)
31, 2syl3an1 1163 . . . 4 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ 𝐹 ∈ V)
4 f1stres 7995 . . . . 5 (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})):((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})⟢(𝐡 βˆ– ran 𝐹)
5 difexg 5326 . . . . . . 7 (𝐡 ∈ π‘Š β†’ (𝐡 βˆ– ran 𝐹) ∈ V)
653ad2ant3 1135 . . . . . 6 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (𝐡 βˆ– ran 𝐹) ∈ V)
7 snex 5430 . . . . . 6 {𝒫 βˆͺ ran 𝐴} ∈ V
8 xpexg 7733 . . . . . 6 (((𝐡 βˆ– ran 𝐹) ∈ V ∧ {𝒫 βˆͺ ran 𝐴} ∈ V) β†’ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}) ∈ V)
96, 7, 8sylancl 586 . . . . 5 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}) ∈ V)
10 fex2 7920 . . . . 5 (((1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})):((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})⟢(𝐡 βˆ– ran 𝐹) ∧ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}) ∈ V ∧ (𝐡 βˆ– ran 𝐹) ∈ V) β†’ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) ∈ V)
114, 9, 6, 10mp3an2i 1466 . . . 4 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) ∈ V)
12 unexg 7732 . . . 4 ((𝐹 ∈ V ∧ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) ∈ V) β†’ (𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) ∈ V)
133, 11, 12syl2anc 584 . . 3 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) ∈ V)
14 cnvexg 7911 . . 3 ((𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) ∈ V β†’ β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) ∈ V)
1513, 14syl 17 . 2 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) ∈ V)
16 eqid 2732 . . . . . . 7 β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) = β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})))
1716domss2 9132 . . . . . 6 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))):𝐡–1-1-ontoβ†’ran β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) ∧ 𝐴 βŠ† ran β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) ∧ (β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) ∘ 𝐹) = ( I β†Ύ 𝐴)))
1817simp1d 1142 . . . . 5 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))):𝐡–1-1-ontoβ†’ran β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))))
19 f1of1 6829 . . . . 5 (β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))):𝐡–1-1-ontoβ†’ran β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) β†’ β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))):𝐡–1-1β†’ran β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))))
2018, 19syl 17 . . . 4 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))):𝐡–1-1β†’ran β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))))
21 ssv 4005 . . . 4 ran β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) βŠ† V
22 f1ss 6790 . . . 4 ((β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))):𝐡–1-1β†’ran β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) ∧ ran β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) βŠ† V) β†’ β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))):𝐡–1-1β†’V)
2320, 21, 22sylancl 586 . . 3 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))):𝐡–1-1β†’V)
2417simp3d 1144 . . 3 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) ∘ 𝐹) = ( I β†Ύ 𝐴))
2523, 24jca 512 . 2 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))):𝐡–1-1β†’V ∧ (β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) ∘ 𝐹) = ( I β†Ύ 𝐴)))
26 f1eq1 6779 . . 3 (𝑔 = β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) β†’ (𝑔:𝐡–1-1β†’V ↔ β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))):𝐡–1-1β†’V))
27 coeq1 5855 . . . 4 (𝑔 = β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) β†’ (𝑔 ∘ 𝐹) = (β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) ∘ 𝐹))
2827eqeq1d 2734 . . 3 (𝑔 = β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) β†’ ((𝑔 ∘ 𝐹) = ( I β†Ύ 𝐴) ↔ (β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) ∘ 𝐹) = ( I β†Ύ 𝐴)))
2926, 28anbi12d 631 . 2 (𝑔 = β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) β†’ ((𝑔:𝐡–1-1β†’V ∧ (𝑔 ∘ 𝐹) = ( I β†Ύ 𝐴)) ↔ (β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))):𝐡–1-1β†’V ∧ (β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) ∘ 𝐹) = ( I β†Ύ 𝐴))))
3015, 25, 29spcedv 3588 1 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ βˆƒπ‘”(𝑔:𝐡–1-1β†’V ∧ (𝑔 ∘ 𝐹) = ( I β†Ύ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  Vcvv 3474   βˆ– cdif 3944   βˆͺ cun 3945   βŠ† wss 3947  π’« cpw 4601  {csn 4627  βˆͺ cuni 4907   I cid 5572   Γ— cxp 5673  β—‘ccnv 5674  ran crn 5676   β†Ύ cres 5677   ∘ ccom 5679  βŸΆwf 6536  β€“1-1β†’wf1 6537  β€“1-1-ontoβ†’wf1o 6539  1st c1st 7969
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-1st 7971  df-2nd 7972  df-en 8936
This theorem is referenced by: (None)
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