Theorem funfocofob 44570

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 6632	. . . . . . . 8 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹)
2	1	biimpi 215	. . . . . . 7 ⊢ (Fun 𝐹 → 𝐹:dom 𝐹⟶ran 𝐹)
3	2	3ad2ant1 1132	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → 𝐹:dom 𝐹⟶ran 𝐹)
4	3	adantr 481	. . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) ∧ (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵) → 𝐹:dom 𝐹⟶ran 𝐹)
5		eqid 2738	. . . . 5 ⊢ (ran 𝐹 ∩ 𝐴) = (ran 𝐹 ∩ 𝐴)
6		eqid 2738	. . . . 5 ⊢ (◡𝐹 “ 𝐴) = (◡𝐹 “ 𝐴)
7		eqid 2738	. . . . 5 ⊢ (𝐹 ↾ (◡𝐹 “ 𝐴)) = (𝐹 ↾ (◡𝐹 “ 𝐴))
8		simp2 1136	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → 𝐺:𝐴⟶𝐵)
9	8	adantr 481	. . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) ∧ (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵) → 𝐺:𝐴⟶𝐵)
10		eqid 2738	. . . . 5 ⊢ (𝐺 ↾ (ran 𝐹 ∩ 𝐴)) = (𝐺 ↾ (ran 𝐹 ∩ 𝐴))
11		simpr 485	. . . . 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 44565	. . . 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 413	. . 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 4149	. . . . . . . . 9 ⊢ (𝐴 ⊆ ran 𝐹 ↔ (ran 𝐹 ∩ 𝐴) = 𝐴)
15	14	biimpi 215	. . . . . . . 8 ⊢ (𝐴 ⊆ ran 𝐹 → (ran 𝐹 ∩ 𝐴) = 𝐴)
16	15	3ad2ant3 1134	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → (ran 𝐹 ∩ 𝐴) = 𝐴)
17	8	fdmd 6611	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → dom 𝐺 = 𝐴)
18	16, 17	eqtr4d 2781	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → (ran 𝐹 ∩ 𝐴) = dom 𝐺)
19	18	reseq2d 5891	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → (𝐺 ↾ (ran 𝐹 ∩ 𝐴)) = (𝐺 ↾ dom 𝐺))
20	8	freld 6606	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → Rel 𝐺)
21		resdm 5936	. . . . . 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 2778	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → (𝐺 ↾ (ran 𝐹 ∩ 𝐴)) = 𝐺)
24		eqidd 2739	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → 𝐵 = 𝐵)
25	23, 16, 24	foeq123d 6709	. . 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 238	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → ((𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵 → 𝐺:𝐴–onto→𝐵))
27		simpr 485	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) ∧ 𝐺:𝐴–onto→𝐵) → 𝐺:𝐴–onto→𝐵)
28		simpl1 1190	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) ∧ 𝐺:𝐴–onto→𝐵) → Fun 𝐹)
29		simpl3 1192	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) ∧ 𝐺:𝐴–onto→𝐵) → 𝐴 ⊆ ran 𝐹)
30		focofo 6701	. . . 4 ⊢ ((𝐺:𝐴–onto→𝐵 ∧ Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹) → (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵)
31	27, 28, 29, 30	syl3anc 1370	. . 3 ⊢ (((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) ∧ 𝐺:𝐴–onto→𝐵) → (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵)
32	31	ex 413	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → (𝐺:𝐴–onto→𝐵 → (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵))
33	26, 32	impbid 211	1 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → ((𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵 ↔ 𝐺:𝐴–onto→𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∩ cin 3886   ⊆ wss 3887  ^◡ccnv 5588  dom cdm 5589  ran crn 5590   ↾ cres 5591   “ cima 5592   ∘ ccom 5593  Rel wrel 5594  Fun wfun 6427  ⟶wf 6429  –onto→wfo 6431

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fo 6439  df-fv 6441

This theorem is referenced by:  fnfocofob  44571