Theorem funfocofob 47815

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 6737	. . . . . . . 8 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹)
2	1	biimpi 219	. . . . . . 7 ⊢ (Fun 𝐹 → 𝐹:dom 𝐹⟶ran 𝐹)
3	2	3ad2ant1 1151	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → 𝐹:dom 𝐹⟶ran 𝐹)
4	3	adantr 485	. . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) ∧ (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵) → 𝐹:dom 𝐹⟶ran 𝐹)
5		eqid 2763	. . . . 5 ⊢ (ran 𝐹 ∩ 𝐴) = (ran 𝐹 ∩ 𝐴)
6		eqid 2763	. . . . 5 ⊢ (◡𝐹 “ 𝐴) = (◡𝐹 “ 𝐴)
7		eqid 2763	. . . . 5 ⊢ (𝐹 ↾ (◡𝐹 “ 𝐴)) = (𝐹 ↾ (◡𝐹 “ 𝐴))
8		simp2 1155	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → 𝐺:𝐴⟶𝐵)
9	8	adantr 485	. . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) ∧ (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵) → 𝐺:𝐴⟶𝐵)
10		eqid 2763	. . . . 5 ⊢ (𝐺 ↾ (ran 𝐹 ∩ 𝐴)) = (𝐺 ↾ (ran 𝐹 ∩ 𝐴))
11		simpr 489	. . . . 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 47808	. . . 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 417	. . 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 4176	. . . . . . . . 9 ⊢ (𝐴 ⊆ ran 𝐹 ↔ (ran 𝐹 ∩ 𝐴) = 𝐴)
15	14	biimpi 219	. . . . . . . 8 ⊢ (𝐴 ⊆ ran 𝐹 → (ran 𝐹 ∩ 𝐴) = 𝐴)
16	15	3ad2ant3 1153	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → (ran 𝐹 ∩ 𝐴) = 𝐴)
17	8	fdmd 6716	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → dom 𝐺 = 𝐴)
18	16, 17	eqtr4d 2801	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → (ran 𝐹 ∩ 𝐴) = dom 𝐺)
19	18	reseq2d 5978	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → (𝐺 ↾ (ran 𝐹 ∩ 𝐴)) = (𝐺 ↾ dom 𝐺))
20	8	freld 6712	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → Rel 𝐺)
21		resdm 6025	. . . . . 6 ⊢ (Rel 𝐺 → (𝐺 ↾ dom 𝐺) = 𝐺)
22	20, 21	syl 18	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → (𝐺 ↾ dom 𝐺) = 𝐺)
23	19, 22	eqtrd 2798	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → (𝐺 ↾ (ran 𝐹 ∩ 𝐴)) = 𝐺)
24		eqidd 2764	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → 𝐵 = 𝐵)
25	23, 16, 24	foeq123d 6813	. . 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 489	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) ∧ 𝐺:𝐴–onto→𝐵) → 𝐺:𝐴–onto→𝐵)
28		simpl1 1210	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) ∧ 𝐺:𝐴–onto→𝐵) → Fun 𝐹)
29		simpl3 1212	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) ∧ 𝐺:𝐴–onto→𝐵) → 𝐴 ⊆ ran 𝐹)
30		focofo 6805	. . . 4 ⊢ ((𝐺:𝐴–onto→𝐵 ∧ Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹) → (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵)
31	27, 28, 29, 30	syl3anc 1398	. . 3 ⊢ (((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) ∧ 𝐺:𝐴–onto→𝐵) → (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵)
32	31	ex 417	. 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 400   ∧ w3a 1103   = wceq 1570   ∩ cin 3904   ⊆ wss 3905  ^◡ccnv 5660  dom cdm 5661  ran crn 5662   ↾ cres 5663   “ cima 5664   ∘ ccom 5665  Rel wrel 5666  Fun wfun 6530  ⟶wf 6532  –onto→wfo 6534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544

This theorem is referenced by:  fnfocofob  47816