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Theorem funfocofob 45786
Description: If the domain of a function 𝐺 is a subset of the range of a function 𝐹, then the composition (𝐺 ∘ 𝐹) is surjective iff 𝐺 is surjective. (Contributed by GL and AV, 29-Sep-2024.)
Assertion
Ref Expression
funfocofob ((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) β†’ ((𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐴)–onto→𝐡 ↔ 𝐺:𝐴–onto→𝐡))
Proof of Theorem funfocofob
StepHypRef Expression
1 fdmrn 6750 . . . . . . . 8 (Fun 𝐹 ↔ 𝐹:dom 𝐹⟢ran 𝐹)
21biimpi 215 . . . . . . 7 (Fun 𝐹 β†’ 𝐹:dom 𝐹⟢ran 𝐹)
323ad2ant1 1134 . . . . . 6 ((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) β†’ 𝐹:dom 𝐹⟢ran 𝐹)
43adantr 482 . . . . 5 (((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) ∧ (𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐴)–onto→𝐡) β†’ 𝐹:dom 𝐹⟢ran 𝐹)
5 eqid 2733 . . . . 5 (ran 𝐹 ∩ 𝐴) = (ran 𝐹 ∩ 𝐴)
6 eqid 2733 . . . . 5 (◑𝐹 β€œ 𝐴) = (◑𝐹 β€œ 𝐴)
7 eqid 2733 . . . . 5 (𝐹 β†Ύ (◑𝐹 β€œ 𝐴)) = (𝐹 β†Ύ (◑𝐹 β€œ 𝐴))
8 simp2 1138 . . . . . 6 ((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) β†’ 𝐺:𝐴⟢𝐡)
98adantr 482 . . . . 5 (((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) ∧ (𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐴)–onto→𝐡) β†’ 𝐺:𝐴⟢𝐡)
10 eqid 2733 . . . . 5 (𝐺 β†Ύ (ran 𝐹 ∩ 𝐴)) = (𝐺 β†Ύ (ran 𝐹 ∩ 𝐴))
11 simpr 486 . . . . 5 (((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) ∧ (𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐴)–onto→𝐡) β†’ (𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐴)–onto→𝐡)
124, 5, 6, 7, 9, 10, 11fcoresfo 45781 . . . 4 (((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) ∧ (𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐴)–onto→𝐡) β†’ (𝐺 β†Ύ (ran 𝐹 ∩ 𝐴)):(ran 𝐹 ∩ 𝐴)–onto→𝐡)
1312ex 414 . . 3 ((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) β†’ ((𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐴)–onto→𝐡 β†’ (𝐺 β†Ύ (ran 𝐹 ∩ 𝐴)):(ran 𝐹 ∩ 𝐴)–onto→𝐡))
14 sseqin2 4216 . . . . . . . . 9 (𝐴 βŠ† ran 𝐹 ↔ (ran 𝐹 ∩ 𝐴) = 𝐴)
1514biimpi 215 . . . . . . . 8 (𝐴 βŠ† ran 𝐹 β†’ (ran 𝐹 ∩ 𝐴) = 𝐴)
16153ad2ant3 1136 . . . . . . 7 ((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) β†’ (ran 𝐹 ∩ 𝐴) = 𝐴)
178fdmd 6729 . . . . . . 7 ((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) β†’ dom 𝐺 = 𝐴)
1816, 17eqtr4d 2776 . . . . . 6 ((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) β†’ (ran 𝐹 ∩ 𝐴) = dom 𝐺)
1918reseq2d 5982 . . . . 5 ((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) β†’ (𝐺 β†Ύ (ran 𝐹 ∩ 𝐴)) = (𝐺 β†Ύ dom 𝐺))
208freld 6724 . . . . . 6 ((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) β†’ Rel 𝐺)
21 resdm 6027 . . . . . 6 (Rel 𝐺 β†’ (𝐺 β†Ύ dom 𝐺) = 𝐺)
2220, 21syl 17 . . . . 5 ((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) β†’ (𝐺 β†Ύ dom 𝐺) = 𝐺)
2319, 22eqtrd 2773 . . . 4 ((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) β†’ (𝐺 β†Ύ (ran 𝐹 ∩ 𝐴)) = 𝐺)
24 eqidd 2734 . . . 4 ((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) β†’ 𝐡 = 𝐡)
2523, 16, 24foeq123d 6827 . . 3 ((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) β†’ ((𝐺 β†Ύ (ran 𝐹 ∩ 𝐴)):(ran 𝐹 ∩ 𝐴)–onto→𝐡 ↔ 𝐺:𝐴–onto→𝐡))
2613, 25sylibd 238 . 2 ((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) β†’ ((𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐴)–onto→𝐡 β†’ 𝐺:𝐴–onto→𝐡))
27 simpr 486 . . . 4 (((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) ∧ 𝐺:𝐴–onto→𝐡) β†’ 𝐺:𝐴–onto→𝐡)
28 simpl1 1192 . . . 4 (((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) ∧ 𝐺:𝐴–onto→𝐡) β†’ Fun 𝐹)
29 simpl3 1194 . . . 4 (((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) ∧ 𝐺:𝐴–onto→𝐡) β†’ 𝐴 βŠ† ran 𝐹)
30 focofo 6819 . . . 4 ((𝐺:𝐴–onto→𝐡 ∧ Fun 𝐹 ∧ 𝐴 βŠ† ran 𝐹) β†’ (𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐴)–onto→𝐡)
3127, 28, 29, 30syl3anc 1372 . . 3 (((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) ∧ 𝐺:𝐴–onto→𝐡) β†’ (𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐴)–onto→𝐡)
3231ex 414 . 2 ((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) β†’ (𝐺:𝐴–onto→𝐡 β†’ (𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐴)–onto→𝐡))
3326, 32impbid 211 1 ((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) β†’ ((𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐴)–onto→𝐡 ↔ 𝐺:𝐴–onto→𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∩ cin 3948   βŠ† wss 3949  β—‘ccnv 5676  dom cdm 5677  ran crn 5678   β†Ύ cres 5679   β€œ cima 5680   ∘ ccom 5681  Rel wrel 5682  Fun wfun 6538  βŸΆwf 6540  β€“ontoβ†’wfo 6542
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-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  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-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  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-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552
This theorem is referenced by:  fnfocofob  45787
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