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Theorem foresf1o 31729
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 7938 . . . 4 (𝐴 ∈ 𝑉 β†’ (𝐹:𝐴–onto→𝐡 β†’ 𝐡 ∈ V))
21imp 407 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) β†’ 𝐡 ∈ V)
3 foelrn 7104 . . . . . 6 ((𝐹:𝐴–onto→𝐡 ∧ 𝑦 ∈ 𝐡) β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 = (πΉβ€˜π‘§))
4 fofn 6804 . . . . . . . . . 10 (𝐹:𝐴–onto→𝐡 β†’ 𝐹 Fn 𝐴)
5 eqcom 2739 . . . . . . . . . . 11 ((πΉβ€˜π‘§) = 𝑦 ↔ 𝑦 = (πΉβ€˜π‘§))
6 fniniseg 7058 . . . . . . . . . . . . 13 (𝐹 Fn 𝐴 β†’ (𝑧 ∈ (◑𝐹 β€œ {𝑦}) ↔ (𝑧 ∈ 𝐴 ∧ (πΉβ€˜π‘§) = 𝑦)))
76biimpar 478 . . . . . . . . . . . 12 ((𝐹 Fn 𝐴 ∧ (𝑧 ∈ 𝐴 ∧ (πΉβ€˜π‘§) = 𝑦)) β†’ 𝑧 ∈ (◑𝐹 β€œ {𝑦}))
87anassrs 468 . . . . . . . . . . 11 (((𝐹 Fn 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (πΉβ€˜π‘§) = 𝑦) β†’ 𝑧 ∈ (◑𝐹 β€œ {𝑦}))
95, 8sylan2br 595 . . . . . . . . . 10 (((𝐹 Fn 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (πΉβ€˜π‘§)) β†’ 𝑧 ∈ (◑𝐹 β€œ {𝑦}))
104, 9sylanl1 678 . . . . . . . . 9 (((𝐹:𝐴–onto→𝐡 ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 = (πΉβ€˜π‘§)) β†’ 𝑧 ∈ (◑𝐹 β€œ {𝑦}))
1110ex 413 . . . . . . . 8 ((𝐹:𝐴–onto→𝐡 ∧ 𝑧 ∈ 𝐴) β†’ (𝑦 = (πΉβ€˜π‘§) β†’ 𝑧 ∈ (◑𝐹 β€œ {𝑦})))
1211reximdva 3168 . . . . . . 7 (𝐹:𝐴–onto→𝐡 β†’ (βˆƒπ‘§ ∈ 𝐴 𝑦 = (πΉβ€˜π‘§) β†’ βˆƒπ‘§ ∈ 𝐴 𝑧 ∈ (◑𝐹 β€œ {𝑦})))
1312adantr 481 . . . . . 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 712 . . . 4 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ 𝑦 ∈ 𝐡) β†’ βˆƒπ‘§ ∈ 𝐴 𝑧 ∈ (◑𝐹 β€œ {𝑦}))
1615ralrimiva 3146 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) β†’ βˆ€π‘¦ ∈ 𝐡 βˆƒπ‘§ ∈ 𝐴 𝑧 ∈ (◑𝐹 β€œ {𝑦}))
17 eleq1 2821 . . . 4 (𝑧 = (π‘”β€˜π‘¦) β†’ (𝑧 ∈ (◑𝐹 β€œ {𝑦}) ↔ (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦})))
1817ac6sg 10479 . . 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 6721 . . . . 5 (𝑔:𝐡⟢𝐴 β†’ ran 𝑔 βŠ† 𝐴)
2120ad2antrl 726 . . . 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 3478 . . . . . 6 𝑔 ∈ V
2322rnex 7899 . . . . 5 ran 𝑔 ∈ V
2423elpw 4605 . . . 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 6802 . . . . . 6 (𝐹:𝐴–onto→𝐡 β†’ 𝐹:𝐴⟢𝐡)
2726ad2antlr 725 . . . . 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 6755 . . . 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 6714 . . . . . 6 (𝑔:𝐡⟢𝐴 β†’ 𝑔 Fn 𝐡)
3029ad2antrl 726 . . . . 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 6727 . . . . 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 6907 . . . . . . . 8 (𝑧 ∈ ran 𝑔 β†’ ((𝐹 β†Ύ ran 𝑔)β€˜π‘§) = (πΉβ€˜π‘§))
3433adantl 482 . . . . . . 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 6892 . . . . . 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 1917 . . . . . . . . 9 Ⅎ𝑦(𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡)
37 nfv 1917 . . . . . . . . . 10 Ⅎ𝑦 𝑔:𝐡⟢𝐴
38 nfra1 3281 . . . . . . . . . 10 β„²π‘¦βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦})
3937, 38nfan 1902 . . . . . . . . 9 Ⅎ𝑦(𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))
4036, 39nfan 1902 . . . . . . . 8 Ⅎ𝑦((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐡) ∧ (𝑔:𝐡⟢𝐴 ∧ βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦})))
41 nfv 1917 . . . . . . . 8 Ⅎ𝑦 𝑧 ∈ ran 𝑔
4240, 41nfan 1902 . . . . . . 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 485 . . . . . . . . . . 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 6892 . . . . . . . . . 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 733 . . . . . . . . . . 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 776 . . . . . . . . . . . . 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 724 . . . . . . . . . . . 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 767 . . . . . . . . . . . 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 3245 . . . . . . . . . . . 12 ((βˆ€π‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}) ∧ 𝑦 ∈ 𝐡) β†’ (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}))
5047, 48, 49syl2anc 584 . . . . . . . . . . 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 7058 . . . . . . . . . . . 12 (𝐹 Fn 𝐴 β†’ ((π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦}) ↔ ((π‘”β€˜π‘¦) ∈ 𝐴 ∧ (πΉβ€˜(π‘”β€˜π‘¦)) = 𝑦)))
5251simplbda 500 . . . . . . . . . . 11 ((𝐹 Fn 𝐴 ∧ (π‘”β€˜π‘¦) ∈ (◑𝐹 β€œ {𝑦})) β†’ (πΉβ€˜(π‘”β€˜π‘¦)) = 𝑦)
5345, 50, 52syl2anc 584 . . . . . . . . . 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 2774 . . . . . . . . 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 6892 . . . . . . . 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 2772 . . . . . . 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 6949 . . . . . . . . 9 (𝑔 Fn 𝐡 β†’ (𝑧 ∈ ran 𝑔 ↔ βˆƒπ‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) = 𝑧))
5857biimpa 477 . . . . . . . 8 ((𝑔 Fn 𝐡 ∧ 𝑧 ∈ ran 𝑔) β†’ βˆƒπ‘¦ ∈ 𝐡 (π‘”β€˜π‘¦) = 𝑧)
5930, 58sylan 580 . . . . . . 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 3265 . . . . . 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 2772 . . . . 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 3146 . . . 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 7083 . . . . . . . 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 6907 . . . . . . . 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 729 . . . . . . . 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 776 . . . . . . . . 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 485 . . . . . . . . 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 584 . . . . . . . 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 584 . . . . . . 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 2772 . . . . . 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 413 . . . . 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 3254 . . . 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 7292 . . 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 5974 . . . . 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 2733 . . . . 5 (π‘₯ = ran 𝑔 β†’ 𝐡 = 𝐡)
7875, 76, 77f1oeq123d 6824 . . . 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 3612 . . 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 584 . 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 1938 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 396   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3474   βŠ† wss 3947  π’« cpw 4601  {csn 4627  β—‘ccnv 5674  ran crn 5676   β†Ύ cres 5677   β€œ cima 5678   Fn wfn 6535  βŸΆwf 6536  β€“ontoβ†’wfo 6538  β€“1-1-ontoβ†’wf1o 6539  β€˜cfv 6540
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-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-reg 9583  ax-inf2 9632  ax-ac2 10454
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  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-rmo 3376  df-reu 3377  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-pss 3966  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-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  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-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  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-isom 6549  df-riota 7361  df-ov 7408  df-om 7852  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-en 8936  df-r1 9755  df-rank 9756  df-card 9930  df-ac 10107
This theorem is referenced by:  rabfodom  31730
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