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Theorem foresf1o 31473
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 7893 . . . 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 7061 . . . . . 6 ((𝐹:𝐴–onto→𝐡 ∧ 𝑦 ∈ 𝐡) β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 = (πΉβ€˜π‘§))
4 fofn 6763 . . . . . . . . . 10 (𝐹:𝐴–onto→𝐡 β†’ 𝐹 Fn 𝐴)
5 eqcom 2744 . . . . . . . . . . 11 ((πΉβ€˜π‘§) = 𝑦 ↔ 𝑦 = (πΉβ€˜π‘§))
6 fniniseg 7015 . . . . . . . . . . . . 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 3166 . . . . . . 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 3144 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) β†’ βˆ€π‘¦ ∈ 𝐡 βˆƒπ‘§ ∈ 𝐴 𝑧 ∈ (◑𝐹 β€œ {𝑦}))
17 eleq1 2826 . . . 4 (𝑧 = (π‘”β€˜π‘¦) β†’ (𝑧 ∈ (◑𝐹 β€œ {𝑦}) ↔ (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦})))
1817ac6sg 10431 . . 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 6680 . . . . 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 3452 . . . . . 6 𝑔 ∈ V
2322rnex 7854 . . . . 5 ran 𝑔 ∈ V
2423elpw 4569 . . . 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 6761 . . . . . 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 6714 . . . 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 6673 . . . . . 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 6686 . . . . 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 6866 . . . . . . . 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 6851 . . . . . 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 3270 . . . . . . . . . 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 6851 . . . . . . . . . 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 3234 . . . . . . . . . . . 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 7015 . . . . . . . . . . . 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 2779 . . . . . . . . 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 6851 . . . . . . . 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 2777 . . . . . . 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 6908 . . . . . . . . 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 3254 . . . . . 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 2777 . . . . 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 3144 . . . 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 7040 . . . . . . . 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 6866 . . . . . . . 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 2777 . . . . . 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 3243 . . . 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 7249 . . 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 5937 . . . . 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 2738 . . . . 5 (π‘₯ = ran 𝑔 β†’ 𝐡 = 𝐡)
7875, 76, 77f1oeq123d 6783 . . . 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 3584 . . 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 3065  βˆƒwrex 3074  Vcvv 3448   βŠ† wss 3915  π’« cpw 4565  {csn 4591  β—‘ccnv 5637  ran crn 5639   β†Ύ cres 5640   β€œ cima 5641   Fn wfn 6496  βŸΆwf 6497  β€“ontoβ†’wfo 6499  β€“1-1-ontoβ†’wf1o 6500  β€˜cfv 6501
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-reg 9535  ax-inf2 9584  ax-ac2 10406
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-iin 4962  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-en 8891  df-r1 9707  df-rank 9708  df-card 9882  df-ac 10059
This theorem is referenced by:  rabfodom  31474
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