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Theorem funfocofob 46084
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 6748 . . . . . . . 8 (Fun 𝐹 ↔ 𝐹:dom 𝐹⟢ran 𝐹)
21biimpi 215 . . . . . . 7 (Fun 𝐹 β†’ 𝐹:dom 𝐹⟢ran 𝐹)
323ad2ant1 1131 . . . . . 6 ((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) β†’ 𝐹:dom 𝐹⟢ran 𝐹)
43adantr 479 . . . . 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 2730 . . . . 5 (ran 𝐹 ∩ 𝐴) = (ran 𝐹 ∩ 𝐴)
6 eqid 2730 . . . . 5 (◑𝐹 β€œ 𝐴) = (◑𝐹 β€œ 𝐴)
7 eqid 2730 . . . . 5 (𝐹 β†Ύ (◑𝐹 β€œ 𝐴)) = (𝐹 β†Ύ (◑𝐹 β€œ 𝐴))
8 simp2 1135 . . . . . 6 ((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) β†’ 𝐺:𝐴⟢𝐡)
98adantr 479 . . . . 5 (((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) ∧ (𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐴)–onto→𝐡) β†’ 𝐺:𝐴⟢𝐡)
10 eqid 2730 . . . . 5 (𝐺 β†Ύ (ran 𝐹 ∩ 𝐴)) = (𝐺 β†Ύ (ran 𝐹 ∩ 𝐴))
11 simpr 483 . . . . 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 46079 . . . 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 411 . . 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 4214 . . . . . . . . 9 (𝐴 βŠ† ran 𝐹 ↔ (ran 𝐹 ∩ 𝐴) = 𝐴)
1514biimpi 215 . . . . . . . 8 (𝐴 βŠ† ran 𝐹 β†’ (ran 𝐹 ∩ 𝐴) = 𝐴)
16153ad2ant3 1133 . . . . . . 7 ((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) β†’ (ran 𝐹 ∩ 𝐴) = 𝐴)
178fdmd 6727 . . . . . . 7 ((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) β†’ dom 𝐺 = 𝐴)
1816, 17eqtr4d 2773 . . . . . 6 ((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) β†’ (ran 𝐹 ∩ 𝐴) = dom 𝐺)
1918reseq2d 5980 . . . . 5 ((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) β†’ (𝐺 β†Ύ (ran 𝐹 ∩ 𝐴)) = (𝐺 β†Ύ dom 𝐺))
208freld 6722 . . . . . 6 ((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) β†’ Rel 𝐺)
21 resdm 6025 . . . . . 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 2770 . . . 4 ((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) β†’ (𝐺 β†Ύ (ran 𝐹 ∩ 𝐴)) = 𝐺)
24 eqidd 2731 . . . 4 ((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) β†’ 𝐡 = 𝐡)
2523, 16, 24foeq123d 6825 . . 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 483 . . . 4 (((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) ∧ 𝐺:𝐴–onto→𝐡) β†’ 𝐺:𝐴–onto→𝐡)
28 simpl1 1189 . . . 4 (((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) ∧ 𝐺:𝐴–onto→𝐡) β†’ Fun 𝐹)
29 simpl3 1191 . . . 4 (((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) ∧ 𝐺:𝐴–onto→𝐡) β†’ 𝐴 βŠ† ran 𝐹)
30 focofo 6817 . . . 4 ((𝐺:𝐴–onto→𝐡 ∧ Fun 𝐹 ∧ 𝐴 βŠ† ran 𝐹) β†’ (𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐴)–onto→𝐡)
3127, 28, 29, 30syl3anc 1369 . . 3 (((Fun 𝐹 ∧ 𝐺:𝐴⟢𝐡 ∧ 𝐴 βŠ† ran 𝐹) ∧ 𝐺:𝐴–onto→𝐡) β†’ (𝐺 ∘ 𝐹):(◑𝐹 β€œ 𝐴)–onto→𝐡)
3231ex 411 . 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 394   ∧ w3a 1085   = wceq 1539   ∩ cin 3946   βŠ† wss 3947  β—‘ccnv 5674  dom cdm 5675  ran crn 5676   β†Ύ cres 5677   β€œ cima 5678   ∘ ccom 5679  Rel wrel 5680  Fun wfun 6536  βŸΆwf 6538  β€“ontoβ†’wfo 6540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  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-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  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 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fo 6548  df-fv 6550
This theorem is referenced by:  fnfocofob  46085
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