Theorem funfocofob 44185

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⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → ((𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵 ↔ 𝐺:𝐴–onto→𝐵))

Proof of Theorem funfocofob

Step	Hyp	Ref	Expression
1		fdmrn 6555	. . . . . . . 8 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹)
2	1	biimpi 219	. . . . . . 7 ⊢ (Fun 𝐹 → 𝐹:dom 𝐹⟶ran 𝐹)
3	2	3ad2ant1 1135	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → 𝐹:dom 𝐹⟶ran 𝐹)
4	3	adantr 484	. . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) ∧ (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵) → 𝐹:dom 𝐹⟶ran 𝐹)
5		eqid 2736	. . . . 5 ⊢ (ran 𝐹 ∩ 𝐴) = (ran 𝐹 ∩ 𝐴)
6		eqid 2736	. . . . 5 ⊢ (◡𝐹 “ 𝐴) = (◡𝐹 “ 𝐴)
7		eqid 2736	. . . . 5 ⊢ (𝐹 ↾ (◡𝐹 “ 𝐴)) = (𝐹 ↾ (◡𝐹 “ 𝐴))
8		simp2 1139	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → 𝐺:𝐴⟶𝐵)
9	8	adantr 484	. . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) ∧ (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵) → 𝐺:𝐴⟶𝐵)
10		eqid 2736	. . . . 5 ⊢ (𝐺 ↾ (ran 𝐹 ∩ 𝐴)) = (𝐺 ↾ (ran 𝐹 ∩ 𝐴))
11		simpr 488	. . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) ∧ (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵) → (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵)
12	4, 5, 6, 7, 9, 10, 11	fcoresfo 44180	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) ∧ (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵) → (𝐺 ↾ (ran 𝐹 ∩ 𝐴)):(ran 𝐹 ∩ 𝐴)–onto→𝐵)
13	12	ex 416	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → ((𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵 → (𝐺 ↾ (ran 𝐹 ∩ 𝐴)):(ran 𝐹 ∩ 𝐴)–onto→𝐵))
14		sseqin2 4116	. . . . . . . . 9 ⊢ (𝐴 ⊆ ran 𝐹 ↔ (ran 𝐹 ∩ 𝐴) = 𝐴)
15	14	biimpi 219	. . . . . . . 8 ⊢ (𝐴 ⊆ ran 𝐹 → (ran 𝐹 ∩ 𝐴) = 𝐴)
16	15	3ad2ant3 1137	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → (ran 𝐹 ∩ 𝐴) = 𝐴)
17	8	fdmd 6534	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → dom 𝐺 = 𝐴)
18	16, 17	eqtr4d 2774	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → (ran 𝐹 ∩ 𝐴) = dom 𝐺)
19	18	reseq2d 5836	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → (𝐺 ↾ (ran 𝐹 ∩ 𝐴)) = (𝐺 ↾ dom 𝐺))
20	8	freld 6529	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → Rel 𝐺)
21		resdm 5881	. . . . . 6 ⊢ (Rel 𝐺 → (𝐺 ↾ dom 𝐺) = 𝐺)
22	20, 21	syl 17	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → (𝐺 ↾ dom 𝐺) = 𝐺)
23	19, 22	eqtrd 2771	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → (𝐺 ↾ (ran 𝐹 ∩ 𝐴)) = 𝐺)
24		eqidd 2737	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → 𝐵 = 𝐵)
25	23, 16, 24	foeq123d 6632	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → ((𝐺 ↾ (ran 𝐹 ∩ 𝐴)):(ran 𝐹 ∩ 𝐴)–onto→𝐵 ↔ 𝐺:𝐴–onto→𝐵))
26	13, 25	sylibd 242	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → ((𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵 → 𝐺:𝐴–onto→𝐵))
27		simpr 488	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) ∧ 𝐺:𝐴–onto→𝐵) → 𝐺:𝐴–onto→𝐵)
28		simpl1 1193	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) ∧ 𝐺:𝐴–onto→𝐵) → Fun 𝐹)
29		simpl3 1195	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) ∧ 𝐺:𝐴–onto→𝐵) → 𝐴 ⊆ ran 𝐹)
30		focofo 6624	. . . 4 ⊢ ((𝐺:𝐴–onto→𝐵 ∧ Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹) → (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵)
31	27, 28, 29, 30	syl3anc 1373	. . 3 ⊢ (((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) ∧ 𝐺:𝐴–onto→𝐵) → (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵)
32	31	ex 416	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → (𝐺:𝐴–onto→𝐵 → (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵))
33	26, 32	impbid 215	1 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → ((𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵 ↔ 𝐺:𝐴–onto→𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∩ cin 3852   ⊆ wss 3853  ^◡ccnv 5535  dom cdm 5536  ran crn 5537   ↾ cres 5538   “ cima 5539   ∘ ccom 5540  Rel wrel 5541  Fun wfun 6352  ⟶wf 6354  –onto→wfo 6356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-fo 6364  df-fv 6366

This theorem is referenced by:  fnfocofob  44186