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Theorem foresf1o 31742
Description: From a surjective function, *choose* a subset of the domain, such that the restricted function is bijective. (Contributed by Thierry Arnoux, 27-Jan-2020.)
Assertion
Ref Expression
foresf1o ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) β†’ βˆƒπ‘₯ ∈ 𝒫 𝐴(𝐹 β†Ύ π‘₯):π‘₯–1-1-onto→𝐡)
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐡   π‘₯,𝐹
Allowed substitution hint:   𝑉(π‘₯)
Proof of Theorem foresf1o
Dummy variables 𝑔 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 focdmex 7942 . . . 4 (𝐴 ∈ 𝑉 β†’ (𝐹:𝐴–onto→𝐡 β†’ 𝐡 ∈ V))
21imp 408 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) β†’ 𝐡 ∈ V)
3 foelrn 7108 . . . . . 6 ((𝐹:𝐴–onto→𝐡 ∧ 𝑦 ∈ 𝐡) β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 = (πΉβ€˜π‘§))
4 fofn 6808 . . . . . . . . . 10 (𝐹:𝐴–onto→𝐡 β†’ 𝐹 Fn 𝐴)
5 eqcom 2740 . . . . . . . . . . 11 ((πΉβ€˜π‘§) = 𝑦 ↔ 𝑦 = (πΉβ€˜π‘§))
6 fniniseg 7062 . . . . . . . . . . . . 13 (𝐹 Fn 𝐴 β†’ (𝑧 ∈ (◑𝐹 β€œ {𝑦}) ↔ (𝑧 ∈ 𝐴 ∧ (πΉβ€˜π‘§) = 𝑦)))
76biimpar 479 . . . . . . . . . . . 12 ((𝐹 Fn 𝐴 ∧ (𝑧 ∈ 𝐴 ∧ (πΉβ€˜π‘§) = 𝑦)) β†’ 𝑧 ∈ (◑𝐹 β€œ {𝑦}))
87anassrs 469 . . . . . . . . . . 11 (((𝐹 Fn 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (πΉβ€˜π‘§) = 𝑦) β†’ 𝑧 ∈ (◑𝐹 β€œ {𝑦}))
95, 8sylan2br 596 . . . . . . . . . 10 (((𝐹 Fn 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (πΉβ€˜π‘§)) β†’ 𝑧 ∈ (◑𝐹 β€œ {𝑦}))
104, 9sylanl1 679 . . . . . . . . 9 (((𝐹:𝐴–onto→𝐡 ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (πΉβ€˜π‘§)) β†’ 𝑧 ∈ (◑𝐹 β€œ {𝑦}))
1110ex 414 . . . . . . . 8 ((𝐹:𝐴–onto→𝐡 ∧ 𝑧 ∈ 𝐴) β†’ (𝑦 = (πΉβ€˜π‘§) β†’ 𝑧 ∈ (◑𝐹 β€œ {𝑦})))
1211reximdva 3169 . . . . . . 7 (𝐹:𝐴–onto→𝐡 β†’ (βˆƒπ‘§ ∈ 𝐴 𝑦 = (πΉβ€˜π‘§) β†’ βˆƒπ‘§ ∈ 𝐴 𝑧 ∈ (◑𝐹 β€œ {𝑦})))
1312adantr 482 . . . . . 6 ((𝐹:𝐴–onto→𝐡 ∧ 𝑦 ∈ 𝐡) β†’ (βˆƒπ‘§ ∈ 𝐴 𝑦 = (πΉβ€˜π‘§) β†’ βˆƒπ‘§ ∈ 𝐴 𝑧 ∈ (◑𝐹 β€œ {𝑦})))
143, 13mpd 15 . . . . 5 ((𝐹:𝐴–onto→𝐡 ∧ 𝑦 ∈ 𝐡) β†’ βˆƒπ‘§ ∈ 𝐴 𝑧 ∈ (◑𝐹 β€œ {𝑦}))
1514adantll 713 . . . 4 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ 𝑦 ∈ 𝐡) β†’ βˆƒπ‘§ ∈ 𝐴 𝑧 ∈ (◑𝐹 β€œ {𝑦}))
1615ralrimiva 3147 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) β†’ βˆ€π‘¦ ∈ 𝐡 βˆƒπ‘§ ∈ 𝐴 𝑧 ∈ (◑𝐹 β€œ {𝑦}))
17 eleq1 2822 . . . 4 (𝑧 = (π‘”β€˜π‘¦) β†’ (𝑧 ∈ (◑𝐹 β€œ {𝑦}) ↔ (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦})))
1817ac6sg 10483 . . 3 (𝐡 ∈ V β†’ (βˆ€π‘¦ ∈ 𝐡 βˆƒπ‘§ ∈ 𝐴 𝑧 ∈ (◑𝐹 β€œ {𝑦}) β†’ βˆƒπ‘”(𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))))
192, 16, 18sylc 65 . 2 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) β†’ βˆƒπ‘”(𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦})))
20 frn 6725 . . . . 5 (𝑔:𝐡⟢𝐴 β†’ ran 𝑔 βŠ† 𝐴)
2120ad2antrl 727 . . . 4 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) β†’ ran 𝑔 βŠ† 𝐴)
22 vex 3479 . . . . . 6 𝑔 ∈ V
2322rnex 7903 . . . . 5 ran 𝑔 ∈ V
2423elpw 4607 . . . 4 (ran 𝑔 ∈ 𝒫 𝐴 ↔ ran 𝑔 βŠ† 𝐴)
2521, 24sylibr 233 . . 3 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) β†’ ran 𝑔 ∈ 𝒫 𝐴)
26 fof 6806 . . . . . 6 (𝐹:𝐴–onto→𝐡 β†’ 𝐹:𝐴⟢𝐡)
2726ad2antlr 726 . . . . 5 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) β†’ 𝐹:𝐴⟢𝐡)
2827, 21fssresd 6759 . . . 4 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) β†’ (𝐹 β†Ύ ran 𝑔):ran π‘”βŸΆπ΅)
29 ffn 6718 . . . . . 6 (𝑔:𝐡⟢𝐴 β†’ 𝑔 Fn 𝐡)
3029ad2antrl 727 . . . . 5 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) β†’ 𝑔 Fn 𝐡)
31 dffn3 6731 . . . . 5 (𝑔 Fn 𝐡 ↔ 𝑔:𝐡⟢ran 𝑔)
3230, 31sylib 217 . . . 4 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) β†’ 𝑔:𝐡⟢ran 𝑔)
33 fvres 6911 . . . . . . . 8 (𝑧 ∈ ran 𝑔 β†’ ((𝐹 β†Ύ ran 𝑔)β€˜π‘§) = (πΉβ€˜π‘§))
3433adantl 483 . . . . . . 7 ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) β†’ ((𝐹 β†Ύ ran 𝑔)β€˜π‘§) = (πΉβ€˜π‘§))
3534fveq2d 6896 . . . . . 6 ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) β†’ (π‘”β€˜((𝐹 β†Ύ ran 𝑔)β€˜π‘§)) = (π‘”β€˜(πΉβ€˜π‘§)))
36 nfv 1918 . . . . . . . . 9 Ⅎ𝑦(𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡)
37 nfv 1918 . . . . . . . . . 10 Ⅎ𝑦 𝑔:𝐡⟢𝐴
38 nfra1 3282 . . . . . . . . . 10 β„²π‘¦βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦})
3937, 38nfan 1903 . . . . . . . . 9 Ⅎ𝑦(𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))
4036, 39nfan 1903 . . . . . . . 8 Ⅎ𝑦((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦})))
41 nfv 1918 . . . . . . . 8 Ⅎ𝑦 𝑧 ∈ ran 𝑔
4240, 41nfan 1903 . . . . . . 7 Ⅎ𝑦(((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔)
43 simpr 486 . . . . . . . . . . 11 ((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦 ∈ 𝐡) ∧ (π‘”β€˜π‘¦) = 𝑧) β†’ (π‘”β€˜π‘¦) = 𝑧)
4443fveq2d 6896 . . . . . . . . . 10 ((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦 ∈ 𝐡) ∧ (π‘”β€˜π‘¦) = 𝑧) β†’ (πΉβ€˜(π‘”β€˜π‘¦)) = (πΉβ€˜π‘§))
454ad5antlr 734 . . . . . . . . . . 11 ((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦 ∈ 𝐡) ∧ (π‘”β€˜π‘¦) = 𝑧) β†’ 𝐹 Fn 𝐴)
46 simplrr 777 . . . . . . . . . . . . 13 ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) β†’ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))
4746ad2antrr 725 . . . . . . . . . . . 12 ((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦 ∈ 𝐡) ∧ (π‘”β€˜π‘¦) = 𝑧) β†’ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))
48 simplr 768 . . . . . . . . . . . 12 ((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦 ∈ 𝐡) ∧ (π‘”β€˜π‘¦) = 𝑧) β†’ 𝑦 ∈ 𝐡)
49 rspa 3246 . . . . . . . . . . . 12 ((βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}) ∧ 𝑦 ∈ 𝐡) β†’ (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))
5047, 48, 49syl2anc 585 . . . . . . . . . . 11 ((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦 ∈ 𝐡) ∧ (π‘”β€˜π‘¦) = 𝑧) β†’ (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))
51 fniniseg 7062 . . . . . . . . . . . 12 (𝐹 Fn 𝐴 β†’ ((π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}) ↔ ((π‘”β€˜π‘¦) ∈ 𝐴 ∧ (πΉβ€˜(π‘”β€˜π‘¦)) = 𝑦)))
5251simplbda 501 . . . . . . . . . . 11 ((𝐹 Fn 𝐴 ∧ (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦})) β†’ (πΉβ€˜(π‘”β€˜π‘¦)) = 𝑦)
5345, 50, 52syl2anc 585 . . . . . . . . . 10 ((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦 ∈ 𝐡) ∧ (π‘”β€˜π‘¦) = 𝑧) β†’ (πΉβ€˜(π‘”β€˜π‘¦)) = 𝑦)
5444, 53eqtr3d 2775 . . . . . . . . 9 ((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦 ∈ 𝐡) ∧ (π‘”β€˜π‘¦) = 𝑧) β†’ (πΉβ€˜π‘§) = 𝑦)
5554fveq2d 6896 . . . . . . . 8 ((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦 ∈ 𝐡) ∧ (π‘”β€˜π‘¦) = 𝑧) β†’ (π‘”β€˜(πΉβ€˜π‘§)) = (π‘”β€˜π‘¦))
5655, 43eqtrd 2773 . . . . . . 7 ((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) ∧ 𝑦 ∈ 𝐡) ∧ (π‘”β€˜π‘¦) = 𝑧) β†’ (π‘”β€˜(πΉβ€˜π‘§)) = 𝑧)
57 fvelrnb 6953 . . . . . . . . 9 (𝑔 Fn 𝐡 β†’ (𝑧 ∈ ran 𝑔 ↔ βˆƒπ‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) = 𝑧))
5857biimpa 478 . . . . . . . 8 ((𝑔 Fn 𝐡 ∧ 𝑧 ∈ ran 𝑔) β†’ βˆƒπ‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) = 𝑧)
5930, 58sylan 581 . . . . . . 7 ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) β†’ βˆƒπ‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) = 𝑧)
6042, 56, 59r19.29af 3266 . . . . . 6 ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) β†’ (π‘”β€˜(πΉβ€˜π‘§)) = 𝑧)
6135, 60eqtrd 2773 . . . . 5 ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) ∧ 𝑧 ∈ ran 𝑔) β†’ (π‘”β€˜((𝐹 β†Ύ ran 𝑔)β€˜π‘§)) = 𝑧)
6261ralrimiva 3147 . . . 4 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) β†’ βˆ€π‘§ ∈ ran 𝑔(π‘”β€˜((𝐹 β†Ύ ran 𝑔)β€˜π‘§)) = 𝑧)
6332ffvelcdmda 7087 . . . . . . . 8 ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) ∧ 𝑦 ∈ 𝐡) β†’ (π‘”β€˜π‘¦) ∈ ran 𝑔)
64 fvres 6911 . . . . . . . 8 ((π‘”β€˜π‘¦) ∈ ran 𝑔 β†’ ((𝐹 β†Ύ ran 𝑔)β€˜(π‘”β€˜π‘¦)) = (πΉβ€˜(π‘”β€˜π‘¦)))
6563, 64syl 17 . . . . . . 7 ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) ∧ 𝑦 ∈ 𝐡) β†’ ((𝐹 β†Ύ ran 𝑔)β€˜(π‘”β€˜π‘¦)) = (πΉβ€˜(π‘”β€˜π‘¦)))
664ad3antlr 730 . . . . . . . 8 ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) ∧ 𝑦 ∈ 𝐡) β†’ 𝐹 Fn 𝐴)
67 simplrr 777 . . . . . . . . 9 ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) ∧ 𝑦 ∈ 𝐡) β†’ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))
68 simpr 486 . . . . . . . . 9 ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) ∧ 𝑦 ∈ 𝐡) β†’ 𝑦 ∈ 𝐡)
6967, 68, 49syl2anc 585 . . . . . . . 8 ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) ∧ 𝑦 ∈ 𝐡) β†’ (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))
7066, 69, 52syl2anc 585 . . . . . . 7 ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) ∧ 𝑦 ∈ 𝐡) β†’ (πΉβ€˜(π‘”β€˜π‘¦)) = 𝑦)
7165, 70eqtrd 2773 . . . . . 6 ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) ∧ 𝑦 ∈ 𝐡) β†’ ((𝐹 β†Ύ ran 𝑔)β€˜(π‘”β€˜π‘¦)) = 𝑦)
7271ex 414 . . . . 5 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) β†’ (𝑦 ∈ 𝐡 β†’ ((𝐹 β†Ύ ran 𝑔)β€˜(π‘”β€˜π‘¦)) = 𝑦))
7340, 72ralrimi 3255 . . . 4 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) β†’ βˆ€π‘¦ ∈ 𝐡 ((𝐹 β†Ύ ran 𝑔)β€˜(π‘”β€˜π‘¦)) = 𝑦)
7428, 32, 62, 732fvidf1od 7296 . . 3 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) β†’ (𝐹 β†Ύ ran 𝑔):ran 𝑔–1-1-onto→𝐡)
75 reseq2 5977 . . . . 5 (π‘₯ = ran 𝑔 β†’ (𝐹 β†Ύ π‘₯) = (𝐹 β†Ύ ran 𝑔))
76 id 22 . . . . 5 (π‘₯ = ran 𝑔 β†’ π‘₯ = ran 𝑔)
77 eqidd 2734 . . . . 5 (π‘₯ = ran 𝑔 β†’ 𝐡 = 𝐡)
7875, 76, 77f1oeq123d 6828 . . . 4 (π‘₯ = ran 𝑔 β†’ ((𝐹 β†Ύ π‘₯):π‘₯–1-1-onto→𝐡 ↔ (𝐹 β†Ύ ran 𝑔):ran 𝑔–1-1-onto→𝐡))
7978rspcev 3613 . . 3 ((ran 𝑔 ∈ 𝒫 𝐴 ∧ (𝐹 β†Ύ ran 𝑔):ran 𝑔–1-1-onto→𝐡) β†’ βˆƒπ‘₯ ∈ 𝒫 𝐴(𝐹 β†Ύ π‘₯):π‘₯–1-1-onto→𝐡)
8025, 74, 79syl2anc 585 . 2 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))) β†’ βˆƒπ‘₯ ∈ 𝒫 𝐴(𝐹 β†Ύ π‘₯):π‘₯–1-1-onto→𝐡)
8119, 80exlimddv 1939 1 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) β†’ βˆƒπ‘₯ ∈ 𝒫 𝐴(𝐹 β†Ύ π‘₯):π‘₯–1-1-onto→𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071  Vcvv 3475   βŠ† wss 3949  π’« cpw 4603  {csn 4629  β—‘ccnv 5676  ran crn 5678   β†Ύ cres 5679   β€œ cima 5680   Fn wfn 6539  βŸΆwf 6540  β€“ontoβ†’wfo 6542  β€“1-1-ontoβ†’wf1o 6543  β€˜cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-reg 9587  ax-inf2 9636  ax-ac2 10458
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-en 8940  df-r1 9759  df-rank 9760  df-card 9934  df-ac 10111
This theorem is referenced by:  rabfodom  31743
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